feat(site/src): move chats between projects from the chat menu - #30895

The chat menu in the sidebar, the top bar and the project page gets a Move to project submenu. It lists the projects of the chat's organization, marks the current one with a check mark and aria-current, and ends with Remove from project.

Stack (moving chats between projects, bottom first):

1. feat: move a chat family between projects in chatd #30893: the move in chatd
2. feat: move chats between projects through the chat update API #30894: the chat update API takes project_id
3. This PR: Move to project in the chat menu
4. feat(site/src/pages/AgentsPage/components): drag chats onto project folders in the sidebar #30795: drag a chat onto a project folder in the sidebar

Why: this is the first way for a user to move a chat, through the API from #30894.

• The submenu shows only for the owner of a root chat that is not archived, with the chat-projects experiment. While the chat or a subagent runs, it looks disabled and does not open, as the API would return 409. It stays keyboard focusable, so keyboard and screen reader users reach the Archive agent hint it names.
• moveChatToProject updates the cached lists and the chat entity at once. Its rollback restores only project_id, so it keeps the fields that watch events wrote while the request ran. A refused move shows the server message in a toast, even after the menu closes.
• Other tabs follow the project_change watch event. A move keeps updated_at, so project_id comes only from that event, and the open chat refetches so the last event cannot win over the committed project. First list fetches that started before the move restart.
• The project page hides a chat that moved away. The composer's project picker no longer says "Can't move to a project later".

What to scrutinize: the project_change branch in AgentsPageLayout.tsx and mergeWatchedChatSummary() in api/queries/chats.ts.

Move to project submenu	Move is off while the chat runs	Project page after the move

Pixel: new ChatTopBar submenu stories, ProjectPage ChatMovedToAnotherProject, and the ChatsSidebar menus without Move to project (archived, shared, subagent) and MoveToProjectOffWhileSubagentRuns, which focuses the blocked item. The open CompactProjectSelector stories lose the hint.

Testing

• make pre-commit passes. Vitest passes on the 5 touched specs (437 tests), and the Storybook tests of the 4 touched story files pass (134).
• The open chat in AgentsPageLayout.test.tsx ends on the older project when two moves arrive in reverse order and it does not refetch. The resetUnloadedChatLists tests fail when the project list family or the unloaded-only check is removed.
• A blocked Move to project takes keyboard focus, has the hint as its description and does not open. The test fails when the trigger is disabled or the open guard is removed.
• Untested: aria-current on the current project, which site/AGENTS.md leaves to the stories.
• Codex: the keyboard reach of a blocked Move to project is fixed. Seven other findings are declined in their threads: client race and cache sync hardening, and story and doc nits on patterns or text this PR keeps from main.

Xum opened this PR on behalf of @ibetitsmike.