Agentic UI: fix resizer for chat input

[nightnei]

Related issues

• Fixes STU-2101

How AI was used in this PR

AI assisted

Proposed Changes

Shaun asked to avoid highlighting the chat input resizer on hover, so this PR addresses it.
@shaunandrews I was thinking about accessing the resizer via tab button, sicne we need to highlight it in this case, so I left the highlight via tab. Is it ok or ytou would you prefer another approach?

Testing Instructions

1. Run Agentic UI
2. Click Tab button a few times to have the chat input resizer focused
3. Assert that it's highlighted, meaning no regressions:
   
4. Mouse click anywhere inside Studio to lose focus
5. Try to resize the chat input
6. Assert that in this case it's not highlighted

[shaunandrews]

Yup! Makes a lot of sense for keyboard resizing. Can we make the line rounded on the ends, similar to how the handle looks for the sidebar.

[wpmobilebot]

📊 Performance Test Results

Comparing 42fbe1c vs trunk

app-size

Metric	trunk	42fbe1c	Diff	Change
App Size (Mac)	1375.46 MB	1381.97 MB	+6.51 MB	🔴 0.5%

site-editor

Metric	trunk	42fbe1c	Diff	Change
load	734 ms	1102 ms	+368 ms	🔴 50.1%

site-startup

Metric	trunk	42fbe1c	Diff	Change
siteCreation	6490 ms	6520 ms	+30 ms	⚪ 0.0%
siteStartup	2397 ms	2389 ms	8 ms	⚪ 0.0%

---

Results are median values from multiple test runs.

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

[nightnei]

Yup! Makes a lot of sense for keyboard resizing. Can we make the line rounded on the ends, similar to how the handle looks for the sidebar.

Addressed

[bcotrim]

LGTM 👍