Studio: Tweak system prompt to mention database

[katinthehatsite]

Related issues

Fixes STU-1924

How AI was used in this PR

It was used to make the adjustments to the system prompt.

Proposed Changes

This PR clarifies in system prompt how the user can access the database and how they can view it. Before the changes, this is what Studio Code was returning to me:

After modifying the system prompt:

Testing Instructions

• Pull the changes from this branch
• Start Studio with npm start
• Navigate to Studio Code on one of the sites and ask different questions about database
• Confirm that it gives correct information
• Confirm that it sends you to Overview tab where you can access phpMyAdmin

Pre-merge Checklist

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

[wpmobilebot]

📊 Performance Test Results

Comparing 94838c3 vs trunk

app-size

Metric	trunk	94838c3	Diff	Change
App Size (Mac)	1311.91 MB	1311.91 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	94838c3	Diff	Change
load	1077 ms	1066 ms	11 ms	⚪ 0.0%

site-startup

Metric	trunk	94838c3	Diff	Change
siteCreation	6526 ms	6493 ms	33 ms	⚪ 0.0%
siteStartup	6476 ms	6992 ms	+516 ms	🔴 8.0%

---

Results are median values from multiple test runs.

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

[gavande1]

LGTM!

[katinthehatsite]

Thanks for the review @gavande1 👍