extension_ci: Bump update action and fix cache issues (#51716)

This PR bumps the update action again. It also fixes a bug where a cache
issue could occur during the extension auto bump and adds update
auto-merge for PRs from staff-members in zed-industries/extensions.

Release Notes:

- N/A
6a06f72f18Finn Evers committed on 3/16/2026, 10:26:56 PM· committed by GitHubparent9c3aa4b
2 files changedLine totals unavailable