Fix /me/sites filtering and environment labels, keeping Pressable sites

[bcotrim]

Related issues

• Related to Agentic UI: Site Creation #3941 — extracted from it per review feedback so the filtering/label changes land with focused testing. Re-applies 756ed7c and 12c975a by @shaunandrews with a corrected filters value.

How AI was used in this PR

AI (Claude) extracted the change from #3941, traced the Pressable regression, and validated the corrected filters value against Calypso/Jetpack Cloud source; lint, typecheck, and sync tests were run locally. Manually reviewed.

Proposed Changes

• Our /me/sites requests have always sent filter= (singular), which the API silently ignores — no server-side filtering ever happened, and non-syncable site types were only weeded out client-side.
• This switches to the real filters=jetpack,atomic,wpcom param — the same set Jetpack Cloud sends in production. Including jetpack is what keeps Pressable (Jetpack-connected) sites in every picker; the earlier filters=atomic,wpcom attempt in Agentic UI: Site Creation #3941 dropped them, which is the regression that got it pulled out of that PR.
• Pressable development environments are no longer treated as staging — they now show their own "Development" label instead of "Staging" in the default UI pickers.
• Sites with environment_type=staging are now labeled Staging even when their parent production site isn't in the fetched list (previously they showed as Production).

Testing Instructions

• Sign in with an account that has a Pressable site plus wpcom simple/Atomic sites.
• Default UI → site → Sync: verify the Pressable site still appears and is connectable (this is the key regression check vs the atomic,wpcom value).
• Verify a Pressable development environment shows a "Development" badge (not "Staging"), and staging sites show "Staging".
• CLI: studio pull / studio push against the Pressable site still resolve it.
• Regression: A8C-owned and deleted sites remain excluded from pickers.

Pre-merge Checklist

• Have you checked for TypeScript, React or other console errors?
• Manually verified with a Pressable-connected account (see Testing Instructions).

[wpmobilebot]

📊 Performance Test Results

Comparing 140803e vs trunk

app-size

Metric	trunk	140803e	Diff	Change
App Size (Mac)	1317.17 MB	1317.17 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	140803e	Diff	Change
load	767 ms	1113 ms	+346 ms	🔴 45.1%

site-startup

Metric	trunk	140803e	Diff	Change
siteCreation	6529 ms	6496 ms	33 ms	⚪ 0.0%
siteStartup	1864 ms	1859 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)