chat: remember last model

docs/api/rest.md

@@ -111,6 +111,7 @@ Behavior notes:
 - If `chatId` is present, server validates chat existence.
 - For `chatId` calls, server stores only *new* non-assistant messages from provided history to avoid duplicates.
 - Server persists final assistant output and call metadata (`LlmCall`) in DB.
+- Server updates chat-level model metadata on each call: `lastUsedProvider`/`lastUsedModel`; first successful/failed call also initializes `initiatedProvider`/`initiatedModel` if unset.
 
 ## Searches
 
@@ -151,7 +152,16 @@ Search run notes:
 
 `ChatSummary`
 ```json
-{ "id": "...", "title": null, "createdAt": "...", "updatedAt": "..." }
+{
+  "id": "...",
+  "title": null,
+  "createdAt": "...",
+  "updatedAt": "...",
+  "initiatedProvider": "openai|anthropic|xai|null",
+  "initiatedModel": "string|null",
+  "lastUsedProvider": "openai|anthropic|xai|null",
+  "lastUsedModel": "string|null"
+}
 ```
 
 `Message`
@@ -172,6 +182,10 @@ Search run notes:
   "title": null,
   "createdAt": "...",
   "updatedAt": "...",
+  "initiatedProvider": "openai|anthropic|xai|null",
+  "initiatedModel": "string|null",
+  "lastUsedProvider": "openai|anthropic|xai|null",
+  "lastUsedModel": "string|null",
   "messages": [Message]
 }
 ```