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/workspaceagents_active_chat_internal_test.go

@@ -37,6 +37,7 @@ func TestActiveAgentChatDefinitionsAgree(t *testing.T) {
 			chat, err := db.InsertChat(ctx, database.InsertChatParams{
 				OrganizationID:    org.ID,
 				Status:            status,
+				ClientType:        database.ChatClientTypeUi,
 				OwnerID:           owner.ID,
 				LastModelConfigID: modelConfig.ID,
 				Title:             fmt.Sprintf("%s-archived-%t", status, archived),