Refactor(chat): Introduce custom Chat class for future modifications

- Copied the `Chat` class from `@google/genai` into `packages/server/src/core/geminiChat.ts`.
- This change is in preparation for future modifications to the chat handling logic.
- Updated relevant files to use the new `GeminiChat` class.

Part of https://github.com/google-gemini/gemini-cli/issues/551

packages/server/src/utils/nextSpeakerChecker.ts

@@ -4,8 +4,9 @@
  * SPDX-License-Identifier: Apache-2.0
  */
 
-import { Chat, Content, SchemaUnion, Type } from '@google/genai';
+import { Content, SchemaUnion, Type } from '@google/genai';
 import { GeminiClient } from '../core/client.js';
+import { GeminiChat } from '../core/geminiChat.js';
 
 const CHECK_PROMPT = `Analyze *only* the content and structure of your immediately preceding response (your last turn in the conversation history). Based *strictly* on that response, determine who should logically speak next: the 'user' or the 'model' (you).
 **Decision Rules (apply in order):**
@@ -57,7 +58,7 @@ export interface NextSpeakerResponse {
 }
 
 export async function checkNextSpeaker(
-  chat: Chat,
+  chat: GeminiChat,
   geminiClient: GeminiClient,
 ): Promise<NextSpeakerResponse | null> {
   // We need to capture the curated history because there are many moments when the model will return invalid turns