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/models.go

@@ -1107,6 +1107,64 @@ func AllBuildReasonValues() []BuildReason {
 	}
 }
 
+type ChatClientType string
+
+const (
+	ChatClientTypeUi  ChatClientType = "ui"
+	ChatClientTypeApi ChatClientType = "api"
+)
+
+func (e *ChatClientType) Scan(src interface{}) error {
+	switch s := src.(type) {
+	case []byte:
+		*e = ChatClientType(s)
+	case string:
+		*e = ChatClientType(s)
+	default:
+		return fmt.Errorf("unsupported scan type for ChatClientType: %T", src)
+	}
+	return nil
+}
+
+type NullChatClientType struct {
+	ChatClientType ChatClientType `json:"chat_client_type"`
+	Valid          bool           `json:"valid"` // Valid is true if ChatClientType is not NULL
+}
+
+// Scan implements the Scanner interface.
+func (ns *NullChatClientType) Scan(value interface{}) error {
+	if value == nil {
+		ns.ChatClientType, ns.Valid = "", false
+		return nil
+	}
+	ns.Valid = true
+	return ns.ChatClientType.Scan(value)
+}
+
+// Value implements the driver Valuer interface.
+func (ns NullChatClientType) Value() (driver.Value, error) {
+	if !ns.Valid {
+		return nil, nil
+	}
+	return string(ns.ChatClientType), nil
+}
+
+func (e ChatClientType) Valid() bool {
+	switch e {
+	case ChatClientTypeUi,
+		ChatClientTypeApi:
+		return true
+	}
+	return false
+}
+
+func AllChatClientTypeValues() []ChatClientType {
+	return []ChatClientType{
+		ChatClientTypeUi,
+		ChatClientTypeApi,
+	}
+}
+
 type ChatMessageRole string
 
 const (
@@ -4303,6 +4361,7 @@ type Chat struct {
 	DynamicTools        pqtype.NullRawMessage `db:"dynamic_tools" json:"dynamic_tools"`
 	OrganizationID      uuid.UUID             `db:"organization_id" json:"organization_id"`
 	PlanMode            NullChatPlanMode      `db:"plan_mode" json:"plan_mode"`
+	ClientType          ChatClientType        `db:"client_type" json:"client_type"`
 }
 
 type ChatDebugRun struct {