docs: add authentication and user management API endpoints

Phase 2 part 2: Split auth and user endpoints
- Authentication: register, login, refresh_token, logout
- Users: get_profile, update_profile, update_theme, change_password
- Each endpoint in separate markdown file
- Include request/response examples and error codes

docs/api/users/update_profile.md

@@ -0,0 +1,50 @@
+# Update Profile
+
+Update the current user's profile information.
+
+**Endpoint**: `PUT /api/users/me/profile`
+**Auth**: Required
+**Content-Type**: `application/json`
+
+## Request Body
+
+| Field | Type | Required | Description |
+|--------|------|-----------|-------------|
+| first_name | string | No | User's first name |
+| last_name | string | No | User's last name |
+
+### Example Request
+
+```json
+{
+  "first_name": "John",
+  "last_name": "Smith"
+}
+```
+
+## Response (200 OK)
+
+```json
+{
+  "id": "uuid",
+  "email": "user@example.com",
+  "username": "john",
+  "first_name": "John",
+  "last_name": "Smith",
+  "theme": "tokyo-night",
+  "role": "user",
+  "max_devices": 10,
+  "created_at": "2026-01-31T10:00:00Z"
+}
+```
+
+## Error Responses
+
+| Code | Description |
+|------|-------------|
+| 400 | Invalid input data |
+| 401 | Invalid or expired token |
+
+## Try It Out
+
+<!-- API Explorer will be inserted here in Phase 3 -->