Get Connection Manifest

The org's live external connections, for the chat composer's picker.

Distinct from /status, which reports every integration row and its health so the Connections page can offer reconnects. This returns only what actually RESOLVED for this turn's execution scope, because the composer's job is to offer connections the agent can genuinely act on right now -- a picker that lists a gated or auth-failed provider invites the user to point the agent at tools it does not hold, and the agent then looks like it ignored them.

Google and Slack are absent by design (native requester-scoped tooling, and a surface with no agent tools respectively); the client composes those in.