docs(handoff): apply Rachael's review suggestions and final copy polish

Apply all 9 of @rachaelrenk's suggested edits on PR #85 and run a final
senior-copywriter pass for consistency across the three handoff pages.

index.mdx
- Intro: "your local Warp terminal" -> "a local Warp session" so the
  intro talks about a session rather than the terminal application.
- "Third-party harness coverage" -> "Third-party agent runtime
  coverage"; same swap in the two bullets underneath. "Harness" is
  internal vocabulary; "agent runtime" is the user-facing term.
- "spiraled into work" -> "grown into work" (less idiomatic).
- "Managing Cloud Agents" -> "Managing cloud agents" in Related pages
  to match the sentence-case nav style used elsewhere.
- Consistency polish: "Fork a cloud conversation into your local
  terminal" -> "into a local Warp session" so the cloud-to-local
  bullet uses the same terminology as the rewritten intro.

local-to-cloud.mdx
- Intro: "Hand off a conversation from the Warp terminal" -> "Hand off
  a local Warp conversation" (same reason as above).
- "This direction is the right move when:" -> "Use this handoff
  direction when:" (less conversational, matches docs voice).
- Post-handoff pointer: linked the management view doc -> deep-linked
  the [Runs page](https://oz.warp.dev/runs) in the Oz web app, which
  matches the pattern added in PR #84.
- "Cloud Agents quickstart" -> "Cloud agents quickstart" (sentence
  case).

cloud-to-cloud.mdx (consistency polish, same patterns Rachael called
out in the other two pages)
- "This direction is the right move when:" -> "Use this handoff
  direction when:".
- Step 1 of "Sending a follow-up": linked the management view doc ->
  deep-linked the Runs page in the Oz web app.
- Removed a stray duplicate `### Third-party harnesses` heading that
  was sitting directly above the correct `### Third-party agent
  runtimes` heading, and inserted a missing blank line before the
  next `## Inspecting a run that's been handed off` section.
- "Managing Cloud Agents" -> "Managing cloud agents" in Related pages.

Validation
- python3 .agents/skills/style_lint/style_lint.py --changed -> 0 issues.
- python3 .agents/skills/check_for_broken_links/check_links.py --internal-only
  -> 0 broken links (2,439 internal links across 321 files).

Co-Authored-By: Oz <oz-agent@warp.dev>

Author: hongyi-chen

src/content/docs/agent-platform/cloud-agents/handoff/cloud-to-cloud.mdx

@@ -9,7 +9,7 @@ sidebar:
 
 Hand off a cloud agent run to a fresh cloud session when the original session has ended but you still want to push the same unit of work further. The run continues in the same conversation, with the prior session's workspace state restored, so the agent picks up where it left off instead of starting fresh.
 
-This direction is the right move when:
+Use this handoff direction when:
 
 * A cloud agent finished and you want to send a follow-up after the session has ended.
 * You want to continue an ambient agent run — an agent that runs in the background, such as a scheduled or integration-triggered run — while preserving it as a single unit of work in the [management view](/agent-platform/cloud-agents/managing-cloud-agents/).
@@ -38,15 +38,15 @@ Cloud-to-cloud handoff relies on a snapshot from the prior execution. Older clou
 
 You don't take a separate "hand off" action — you send a follow-up, and the run continues. The handoff happens automatically when the original session has ended. Send-follow-up support is available in the Warp app.
 
-1. **Open the ended cloud run.** Find it in the [management view](/agent-platform/cloud-agents/managing-cloud-agents/) or in the conversation panel.
+1. **Open the ended cloud run.** Find it on the [Runs page](https://oz.warp.dev/runs) in the Oz web app or in the conversation panel.
 2. **Send your follow-up.** Type the next message into the conversation's input and submit it.
 
 The run picks up where it left off, with workspace state restored.
 
-### Third-party harnesses
 ### Third-party agent runtimes
 
 Cloud-to-cloud handoff also works for supported third-party agent runtimes, but the affordance is a **Continue** button followed by an input prompt instead of the standard follow-up input. Click **Continue** on the ended run (from the conversation details panel or the end-of-run tombstone) and enter the next message you want the agent to act on.
+
 ## Inspecting a run that's been handed off
 
 The [management view](/agent-platform/cloud-agents/managing-cloud-agents/) shows one row per run, even when the run spans multiple sessions. Click into the run to see the full transcript: each execution appears in order, and you can scan the transcript to see where one execution ended and the next began. Per-execution timestamps aren't currently exposed in the API; the transcript is the source of truth.
@@ -63,4 +63,4 @@ Very old runs predate the agent conversation model and can't be continued via ha
 
 * [Handoff overview](/agent-platform/cloud-agents/handoff/) - What handoff is, the directions it supports, and what carries over.
 * [Handoff from local to cloud](/agent-platform/cloud-agents/handoff/local-to-cloud/) - Promote a local conversation to a cloud run.
-* [Managing Cloud Agents](/agent-platform/cloud-agents/managing-cloud-agents/) - Find runs to send follow-ups to.
+* [Managing cloud agents](/agent-platform/cloud-agents/managing-cloud-agents/) - Find runs to send follow-ups to.

src/content/docs/agent-platform/cloud-agents/handoff/index.mdx

@@ -8,22 +8,22 @@ sidebar:
   label: "Handoff overview"
 ---
 
-Handoff lets you move a unit of agent work between your local Warp terminal and the cloud, or continue a finished cloud run, without re-explaining the task. The receiving agent picks up the conversation history, your workspace state, and any conversation attachments from the prior session.
+Handoff lets you move a unit of agent work between a local Warp session and the cloud, or continue a finished cloud run, without re-explaining the task. The receiving agent picks up the conversation history, your workspace state, and any conversation attachments from the prior session.
 
 ## Directions of handoff
 
 Handoff supports three directions:
 
 * **Local to cloud** - Promote a local Warp Agent conversation to a cloud agent run when you need more compute, longer-running work, or to fan out variants in parallel. The cloud agent starts from your conversation history and a snapshot of your uncommitted workspace changes. See [Handoff from local to cloud](/agent-platform/cloud-agents/handoff/local-to-cloud/).
 * **Cloud to cloud** - Send a follow-up to a cloud run after its session has ended. The run continues in the same conversation, with the prior session's workspace state restored. See [Handoff from cloud to cloud](/agent-platform/cloud-agents/handoff/cloud-to-cloud/).
-* **Cloud to local** - Fork a cloud conversation into your local terminal with **Continue locally** or `/continue-locally`. See [Viewing cloud agent runs](/agent-platform/cloud-agents/viewing-cloud-agent-runs/#5-fork-the-session-to-your-local-warp).
+* **Cloud to local** - Fork a cloud conversation into a local Warp session with **Continue locally** or `/continue-locally`. See [Viewing cloud agent runs](/agent-platform/cloud-agents/viewing-cloud-agent-runs/#5-fork-the-session-to-your-local-warp).
 
-### Third-party harness coverage
+### Third-party agent runtime coverage
 
 Handoff coverage depends on which agent is running the conversation:
 
-* **Cloud to cloud** works for both the Warp Agent and third-party harnesses. On third-party harnesses, the affordance is a **Continue** button followed by an input prompt, instead of the streamlined follow-up flow you get with the Warp Agent.
-* **Local to cloud** works for the Warp Agent. It isn't available for third-party harnesses.
+* **Cloud to cloud** works for both the Warp Agent and supported third-party agent runtimes. For supported third-party agent runtimes, the affordance is a **Continue** button followed by an input prompt, instead of the streamlined follow-up flow you get with the Warp Agent.
+* **Local to cloud** works for the Warp Agent. It isn't available for supported third-party agent runtimes.
 
 ## What carries over
 
@@ -39,14 +39,14 @@ Handoff is best-effort. When the receiving agent can apply the prior session's c
 
 Each direction has a clear motivating workflow.
 
-* **Local to cloud** - Use when a local conversation has spiraled into work that's better done in the cloud: long-running tasks you don't want to keep your laptop awake for, parallel variants of the same task, or steering work from another device once it's running.
+* **Local to cloud** - Use when a local conversation has grown into work that's better done in the cloud: long-running tasks you don't want to keep your laptop awake for, parallel variants of the same task, or steering work from another device once it's running.
 * **Cloud to cloud** - Use when a cloud agent finished and you want to send a follow-up without losing the prior workspace state. Also useful when a Slack-triggered or scheduled run completes and someone on the team wants to push it further.
 * **Cloud to local** - Use when a cloud agent has done the heavy lifting and you want to take over locally to verify, iterate, or polish before shipping.
 
 ## Related pages
 
 * [Cloud agents overview](/agent-platform/cloud-agents/overview/) - What cloud agents are, when to use them, and how they fit into the Oz Platform.
-* [Managing Cloud Agents](/agent-platform/cloud-agents/managing-cloud-agents/) - Inspect handoff runs from the management view alongside local conversations.
+* [Managing cloud agents](/agent-platform/cloud-agents/managing-cloud-agents/) - Inspect handoff runs from the management view alongside local conversations.
 * [Viewing cloud agent runs](/agent-platform/cloud-agents/viewing-cloud-agent-runs/) - Open and continue a cloud run locally with **Continue locally** or `/continue-locally`.
 * [Cloud-synced conversations](/agent-platform/local-agents/cloud-conversations/) - How conversations sync between local and cloud so handoff can find them.
 * [Environments](/agent-platform/cloud-agents/environments/) - The runtime context a cloud agent runs in after a handoff.

src/content/docs/agent-platform/cloud-agents/handoff/local-to-cloud.mdx

@@ -8,9 +8,9 @@ sidebar:
   label: "Local to cloud"
 ---
 
-Hand off a conversation from the Warp terminal to a cloud agent run when the work outgrows your local session. The cloud agent inherits a fork of your local conversation and a snapshot of your uncommitted workspace changes, so it picks up where you left off instead of starting fresh.
+Hand off a local Warp conversation to a cloud agent run when the work outgrows your local session. The cloud agent inherits a fork of your local conversation and a snapshot of your uncommitted workspace changes, so it picks up where you left off instead of starting fresh.
 
-This direction is the right move when:
+Use this handoff direction when:
 
 * The task is going to run for a long time and you don't want to keep your laptop awake.
 * You want to fan out variants of the same task across multiple cloud agents in parallel.
@@ -46,7 +46,7 @@ The cloud agent runs with the same model your local conversation was using. Chan
 
 After submission, the cloud agent applies your workspace snapshot and addresses your follow-up. The local conversation stays untouched — you can keep working in it locally or close it.
 
-To check on the new run, open it from the [management view](/agent-platform/cloud-agents/managing-cloud-agents/) or the conversation panel.
+To check on the new run, open it from the [Runs page](https://oz.warp.dev/runs) in the Oz web app or the conversation panel.
 
 ## Troubleshooting
 
@@ -64,4 +64,4 @@ The source conversation may not have finished syncing to the cloud when you trig
 * [Handoff overview](/agent-platform/cloud-agents/handoff/) - What handoff is, the directions it supports, and what carries over.
 * [Handoff from cloud to cloud](/agent-platform/cloud-agents/handoff/cloud-to-cloud/) - Continue a finished cloud run with workspace state restored.
 * [Environments](/agent-platform/cloud-agents/environments/) - Configure the repos, image, and setup commands the cloud agent starts in.
-* [Cloud Agents quickstart](/agent-platform/cloud-agents/quickstart/) - Run your first cloud agent from scratch.
+* [Cloud agents quickstart](/agent-platform/cloud-agents/quickstart/) - Run your first cloud agent from scratch.