Add a `chat_client_type` enum (`ui` | `api`) and `client_type` column to
the `chats` table. The column defaults to `api` for new rows so API
callers don't need to set it explicitly. Existing rows are backfilled to
`ui`.
The field flows through `CreateChatRequest`, `chatd.CreateOptions`,
`InsertChat`, and is returned in the `Chat` response via `db2sdk`.
<details>
<summary>Implementation notes (Coder Agents generated)</summary>
### Changes
**Database migration (000469)**
- New enum `chat_client_type` with values `ui`, `api`.
- New `client_type` column, `NOT NULL DEFAULT 'api'`.
- Backfill: `UPDATE chats SET client_type = 'ui'`.
**SQL query** — `InsertChat` now includes `client_type`.
**SDK** — `ChatClientType` type added; `ClientType` field added to both
`CreateChatRequest` (optional, defaults server-side to `api`) and `Chat`
response.
**Handler** — `postChats` maps the request field (defaulting to `api`)
and passes it through `chatd.CreateOptions`.
**Sub-agent** — Child chats inherit their parent's `client_type`.
**db2sdk** — Maps the database value to the SDK type.
### Decision log
- Default is `api` (not `ui`) so existing API integrations get the
correct value without code changes.
- Backfill sets existing rows to `ui` per requirement.
- Child chats inherit `client_type` from parent rather than defaulting.
</details>
Dean Sheather
3452ab3166