Studio: Ensure refresh browser tool has the right icon in the agentic UI

[katinthehatsite]

Related issues

Fixes STU-1983

How AI was used in this PR

Not needed

Proposed Changes

This PR ensures that the refresh_browser tool in the agentic UI uses the refresh icon, similar to the browser refresh:

Testing Instructions

• Pull the changes from this branch
• Start Studio with npm start
• Enable the Agentic UI feature flag from Studio -> Feature flags from the topbar
• Open the chat for one of the sites
• Ask the agent to make a visible change such as to change site title
• Observe the refresh tool appear
• Confirm that it has the correct refresh icon
• Confirm that it looks good both in the dark and light modes

Pre-merge Checklist

• Have you checked for TypeScript, React or other console errors?

[wpmobilebot]

📊 Performance Test Results

Comparing 63eb3d3 vs trunk

app-size

Metric	trunk	63eb3d3	Diff	Change
App Size (Mac)	1419.29 MB	1419.29 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	63eb3d3	Diff	Change
load	1063 ms	1065 ms	+2 ms	⚪ 0.0%

site-startup

Metric	trunk	63eb3d3	Diff	Change
siteCreation	7000 ms	7000 ms	0 ms	⚪ 0.0%
siteStartup	2351 ms	2346 ms	5 ms	⚪ 0.0%

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Results are median values from multiple test runs.

Legend: 🟢 Improvement (faster) | 🔴 Regression (slower) | ⚪ No change (<50ms diff)