## Objective

Fixes an issue where language servers provided by extensions are not
included in the settings schemas served by remote servers. When
connecting to a remote server via the
`open_remote_project_with_existing_connection` flow (dev servers,
multi-workspace reopen, git worktree picker), setup-N servers never
received any extension sync, so their schemas omitted extension-provided
language servers, breaking language support for remote development.

## Solution

Three changes ensure that language servers provided by extensions are
included in the settings schemas served by remote servers:

1. **Register connections opened via
`open_remote_project_with_existing_connection` with the extension
store**: dev servers, multi-workspace reopen, and the git worktree
picker now register their connections with the extension store so that
extensions are synced to those servers. Previously only the
`open_remote_project` flow did this, so setup-N servers never received
any extension sync and their schemas omitted extension language servers.

2. **Emit `ExtensionsInstalledChanged` from `HeadlessExtensionStore`
after sync and install complete**: this allows the remote process to
invalidate its cached settings schemas. The local `ExtensionStore`
already emitted this event; without it, the remote schema cache stayed
stale for the lifetime of the server process.

3. **Include available LSP adapters in the project settings schema**:
this matches the user settings schema (this was fixed for `settings` in
#46766 but never applied to `project_settings`).

## Testing

- Manually verified that after connecting to a remote server via the
`open_remote_project_with_existing_connection` flow (including dev
servers and the git worktree picker), extension language servers appear
in the schema served by the remote server.
- Verified that `ExtensionsInstalledChanged` is emitted after sync and
install complete, and that the remote schema cache is correctly
invalidated and rebuilt.
- Verified that the `project_settings` schema includes the available LSP
adapters, matching the behavior of the user settings schema.
- Reviewers should focus on the multi-workspace reopen and git worktree
picker flows (the two previously unregistered flows) to confirm that
extension language servers are correctly synced and appear in the
schema.

## Self-Review Checklist:

- [x] I've reviewed my own diff for quality, security, and reliability
- [ ] Unsafe blocks (if any) have justifying comments
- [ ] The content adheres to Zed's UI standards
([UX/UI](https://github.com/zed-industries/zed/blob/main/CONTRIBUTING.md#uiux-checklist)
and
[icon](https://github.com/zed-industries/zed/blob/main/crates/icons/README.md)
guidelines)
- [x] Tests cover the new/changed behavior
- [x] Performance impact has been considered and is acceptable

## Showcase
**Server / Project** — before the fix
<img width="1061" height="749" alt="image"
src="https://github.com/user-attachments/assets/d6c5459f-58fb-4769-a701-f6fb317adc37"
/>
**Local**
<img width="1119" height="449" alt="image"
src="https://github.com/user-attachments/assets/fb30a1e2-74f0-4439-a25f-1022ed164074"
/>

---

Release Notes:

- Fixed an issue where language servers provided by extensions were
missing from settings schemas served by remote servers.

---------

Co-authored-by: vancez <vancez@users.noreply.github.com>
Co-authored-by: Kirill Bulatov <kirill@zed.dev>