rust: use web service for Responses sample

Pivot the Rust Responses work away from SDK-native client/types and demonstrate the intended pattern instead: use FoundryLocalManager for setup/model/web-service lifecycle, then call /v1/responses via raw HTTP.

Adds a focused responses_web_service example and integration tests for non-streaming, streaming SSE, and function-calling flows against the local OpenAI-compatible web service.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>

Author: MaanavD

sdk/rust/Cargo.toml

@@ -25,8 +25,6 @@ tokio-stream = "0.1"
 tokio-util = "0.7"
 futures-core = "0.3"
 reqwest = { version = "0.12", features = ["json", "stream"] }
-bytes = "1"
-async-stream = "0.3"
 urlencoding = "2"
 async-openai = { version = "0.33", default-features = false, features = ["chat-completion-types", "embedding-types"] }
 
@@ -37,8 +35,8 @@ serde_json = "1"
 serde = { version = "1", features = ["derive"] }
 
 [[example]]
-name = "responses"
-path = "examples/responses.rs"
+name = "responses_web_service"
+path = "examples/responses_web_service.rs"
 
 [[example]]
 name = "chat_completion"