Studio: Tweak system prompt for push and pull operations

[katinthehatsite]

Related issues

Related to STU-1627

How AI was used in this PR

N/A

Proposed Changes

This PR adds clarification for the Studio Code agent for the pull and push operations to the system prompt. I was testing pulling and pushing sites today and had this interaction with the agent regarding pulling remote sites where it was not clear on requirements and eligibility:

Testing Instructions

• Try pulling a site that is not eligible for a pull, for example, through Studio Code and confirm that the agent has some context about the eligibility of sites for pull

Pre-merge Checklist

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

[Copilot]

Pull request overview

Updates the Studio Code agent’s CLI system prompt to provide clearer guidance for site_push and site_pull operations (syncing local sites with WordPress.com/Pressable), aiming to reduce confusion around sync eligibility and workflows.

Changes:

• Consolidates push guidance under a new “Pull & Push” section and adds eligibility/connection notes.
• Adds pull workflow guidance and a brief troubleshooting section for common sync errors.

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[wpmobilebot]

📊 Performance Test Results

Comparing 9ef58f2 vs trunk

app-size

Metric	trunk	9ef58f2	Diff	Change
App Size (Mac)	2357.79 MB	2357.79 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	9ef58f2	Diff	Change
load	1720 ms	1732 ms	+12 ms	⚪ 0.0%

site-startup

Metric	trunk	9ef58f2	Diff	Change
siteCreation	8572 ms	8498 ms	74 ms	🟢 -0.9%
siteStartup	3917 ms	3870 ms	47 ms	⚪ 0.0%

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Results are median values from multiple test runs.

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

[sejas]

@katinthehatsite , thanks for improving the Pull/Push. I think adding the Pull workflow is a great addition.

I'm not totally sure about the "Troubleshooting" section though. I think it can be useful but we should be careful what we add there and how we measure it helps to the user. Ultimately, I think it should be in a specific skill and not in the system prompt. I suggest removing this section or moving it to a separate skill.

The rest looks good. Thank you!

[sejas]

Is this information available to the Agent?

[katinthehatsite]

I'm not totally sure about the "Troubleshooting" section though. I think it can be useful but we should be careful what we add there and how we measure it helps to the user. Ultimately, I think it should be in a specific skill and not in the system prompt. I suggest removing this section or moving it to a separate skill.

Sounds good to me 👍 I don't have a strong preference for this, just mostly was looking to make the whole push / pull experience smoother for users. I will remove it for now

[sejas]

Thanks Kat! I left one more suggestion.