chore: add client_type field to chats and telemetry (#24342)

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>

docs/ai-coder/agents/chats-api.md

@@ -56,7 +56,8 @@ The response is the newly created `Chat` object:
   "pin_order": 0,
   "mcp_server_ids": [],
   "labels": {},
-  "has_unread": false
+  "has_unread": false,
+  "client_type": "api"
 }
 ```
 
@@ -89,6 +90,7 @@ A typical integration follows three steps:
 | `model_config_id` | `uuid`              | no       | Override the default model configuration.       |
 | `mcp_server_ids`  | `uuid[]`            | no       | Attach MCP servers to this chat.                |
 | `labels`          | `map[string]string` | no       | Key-value labels for the chat (max 50).         |
+| `client_type`     | `string`            | no       | `"ui"` or `"api"`. Defaults to `"api"`.         |
 
 Each `ChatInputPart` has a `type` field. The simplest form is a text part: