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>

coderd/database/queries/chats.sql

@@ -406,7 +406,8 @@ INSERT INTO chats (
     status,
     mcp_server_ids,
     labels,
-    dynamic_tools
+    dynamic_tools,
+    client_type
 ) VALUES (
     @organization_id::uuid,
     @owner_id::uuid,
@@ -422,7 +423,8 @@ INSERT INTO chats (
     @status::chat_status,
     COALESCE(@mcp_server_ids::uuid[], '{}'::uuid[]),
     COALESCE(sqlc.narg('labels')::jsonb, '{}'::jsonb),
-    sqlc.narg('dynamic_tools')::jsonb
+    sqlc.narg('dynamic_tools')::jsonb,
+    @client_type::chat_client_type
 )
 RETURNING
     *;
@@ -1299,7 +1301,7 @@ SELECT
     id, owner_id, created_at, updated_at, status,
     (parent_chat_id IS NOT NULL)::bool AS has_parent,
     root_chat_id, workspace_id,
-    mode, archived, last_model_config_id
+    mode, archived, last_model_config_id, client_type
 FROM chats
 WHERE updated_at > @updated_after;