refactor: rename mascot to chatbot across entire codebase

- Backend: mascot-chat module → chatbot, routes /mascot/chat → /chat
- Shared schema: pgMascotChatMessagesTable → pgChatbotMessagesTable
- Frontend: MascotProvider/useMascot → ChatbotProvider/useChatbot
- Gateway schema: update exports to match shared schema
- Docs: update all .md references (CLAUDE.md, README, specs, plans)
- All API routes, controller names, service classes, types renamed

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

services/backend/src/http/app.ts

@@ -10,7 +10,7 @@ import { createAnalysisRouter } from "../modules/analysis/index.js";
 import { createConfigRouter } from "../modules/config/index.js";
 import { createDashboardRouter } from "../modules/dashboard/index.js";
 import { createHealthRouter } from "../modules/health/index.js";
-import { createMascotChatRouter } from "../modules/mascot-chat/index.js";
+import { createChatbotRouter } from "../modules/chatbot/index.js";
 import { createMediaRouter } from "../modules/media/index.js";
 import { createMessagesRouter } from "../modules/messages/index.js";
 import { createRecordingsRouter } from "../modules/recordings/index.js";
@@ -64,7 +64,7 @@ export function createHttpApp(): Express {
   app.use("/api", createDashboardRouter());
   app.use("/api", createMessagesRouter());
   app.use("/api", createAnalysisRouter());
-  app.use("/api", createMascotChatRouter());
+  app.use("/api", createChatbotRouter());
   app.use("/api", createRecordingsRouter());
   app.use("/api", createUiStateRouter());
   app.use("/api", createMediaRouter());