Fix browse daemon ownership across handoff

[s-ai-alpha]

What changed

• Keep daemon termination ownership tied to the mode it was launched in.
• Prevent a headless daemon promoted by handoff from shutting down when the short-lived launch CLI exits.
• Use the same ownership policy for the parent watchdog and SIGTERM.
• Avoid throwing when Playwright's public Browser object has no process() accessor.

Root cause

The parent watchdog and SIGTERM handler derived lifecycle ownership from the browser's current mode. After handoff changed a headless browser to headed mode, the already-exited one-shot CLI appeared to own the now-visible browser. The next watchdog tick shut the daemon and browser down.

Separately, disconnect handling assumed a Puppeteer-style Browser.process() method that Playwright's public API does not provide.

Impact

Headless-to-headed handoff remains alive until the user closes it or explicit cleanup runs. Ordinary Playwright disconnects no longer throw from the exit-classification path.

Validation

32 pass
0 fail

Covered by:

• browse/test/daemon-ownership-policy.test.ts
• browse/test/watchdog.test.ts
• browse/test/browser-manager-unit.test.ts

The original lifecycle reproduction was also repeated across the 15-second watchdog boundary: the same daemon PID and URL remained healthy after handoff.

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