docs(teams): phase 9 as built — roles, not overwrites, and three things §7.3 named that do not exist

Amends `TEAMS.md` §7.3 inline, marks phase 9 done in Part 12, and adds the
`team_integrations` row to `BACKEND_DESIGN.md`'s schema table. Pairs with
**website#159**.

Co-Authored-By: Claude <noreply@anthropic.com>

website/BACKEND_DESIGN.md

@@ -554,6 +554,7 @@ core's.
 | `team_forum_uploads` | attribution for `uploads` mode: who uploaded what, when, how big, and to which post. Also the sweep's worklist |
 | `team_notification_prefs` | per-Team notification preference (phase 6). **Opt-out for push, opt-IN for email** — `muted` defaults 0 and `email_mode` defaults `'off'`, so the two sinks default opposite ways and the asymmetry lives here rather than in a condition anyone has to remember. Team scoping lives in this table and in the recipient computation, never in a stream id. `last_digest_at` is the digest's only state and the worker is its only writer |
 | `team_integration_config` | where a Team's notifications go on another platform (phase 8). One row per (platform, Team) plus a **deployment-wide default** whose `team_id` is NULL — expressed with a generated `team_key AS IFNULL(team_id, 0)` in the unique key, because a NULL cannot live in a primary key and the default row is the base case of the whole override mechanism. `members_ack` is a **precondition, not a preference**: forum posts and announcements are members-only always, core cannot see a channel's permissions, so enabling one requires an attributed operator acknowledgement that the destination is restricted — and changing the channel clears it |
+| `team_integrations` | a Team's provisioned resource on another platform — today its Discord **voice channel and the role that opens it** (§7.3, phase 9). Both refs on one row because they are one lifecycle: a role for a channel that no longer exists is a badge for nowhere. `state` is core's BELIEF about the platform, never the platform's answer — the reconciler writes what it just did and the next pass re-derives the truth. A Team that stops qualifying goes to `pending_removal` with `remove_after` rather than being deleted at once, so a Team hovering around the size threshold does not delete-and-recreate its channel and change its id. `synced_at` is separate from `updated_at`, which moves whenever core writes a belief including an error |
 | `content_reports` | member-raised abuse reports (phase 5). **Not a `team_*` table and not named for the forum** — `target_type` is a plain VARCHAR so a wiki page or a news comment becomes a value rather than a table. Team forum content is only the first consumer |
 
 **Core had no user-facing report flow of any kind before `content_reports`.** `moderation`,