Preserve temporary message in Studio Code UI

[nightnei]

Related issues

• Fixes STU-1829

How AI was used in this PR

AI wrote the code, I reviewed and made a few manual adjustments.

Proposed Changes

I had a case where I was writing a prompt, then needed to switch to the site Settings tab and come back — and as a result, I lost my prompt draft.

AFAIK, it’s a common UX expectation to preserve drafts when the app state changes or when the user temporarily leaves the form.

Testing Instructions

1. npm start
2. Open Studio Code tab for site A and write "1111" (don't submit it)
3. Open Studio Code tab for site B and write "2222" (don't submit it)
4. Come back to site A and assert that you still see 1111
5. Come back to site B and assert that you still see 2222
6. Try to test it with submitting/clearing the form, switching tabs / coming back, and asserting that everything works well

[wpmobilebot]

📊 Performance Test Results

Comparing 4ff5210 vs trunk

app-size

Metric	trunk	4ff5210	Diff	Change
App Size (Mac)	1354.74 MB	1354.75 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	4ff5210	Diff	Change
load	1810 ms	1748 ms	62 ms	🟢 -3.4%

site-startup

Metric	trunk	4ff5210	Diff	Change
siteCreation	8515 ms	9061 ms	+546 ms	🔴 6.4%
siteStartup	3867 ms	4887 ms	+1020 ms	🔴 26.4%

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Results are median values from multiple test runs.

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