feat(skill-mcp-manager): improve connection handling and client implementations

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

src/features/skill-mcp-manager/manager.ts

@@ -1,4 +1,3 @@
-import type { Client } from "@modelcontextprotocol/sdk/client/index.js"
 import type { Prompt, Resource, Tool } from "@modelcontextprotocol/sdk/types.js"
 import type { ClaudeCodeMcpServer } from "../claude-code-mcp-loader/types"
 import { McpOAuthProvider } from "../mcp-oauth/provider"
@@ -6,6 +5,7 @@ import { disconnectAll, disconnectSession, forceReconnect } from "./cleanup"
 import { getOrCreateClient, getOrCreateClientWithRetryImpl } from "./connection"
 import { handlePostRequestAuthError, handleStepUpIfNeeded } from "./oauth-handler"
 import type {
+  McpClient,
   OAuthProviderFactory,
   SkillMcpClientInfo,
   SkillMcpManagerState,
@@ -36,7 +36,7 @@ export class SkillMcpManager {
     return `${info.sessionID}:${info.skillName}:${info.serverName}`
   }
 
-  async getOrCreateClient(info: SkillMcpClientInfo, config: ClaudeCodeMcpServer): Promise<Client> {
+  async getOrCreateClient(info: SkillMcpClientInfo, config: ClaudeCodeMcpServer): Promise<McpClient> {
     const clientKey = this.getClientKey(info)
     return await getOrCreateClient({
       state: this.state,
@@ -106,7 +106,7 @@ export class SkillMcpManager {
   private async withOperationRetry<T>(
     info: SkillMcpClientInfo,
     config: ClaudeCodeMcpServer,
-    operation: (client: Client) => Promise<T>
+    operation: (client: McpClient) => Promise<T>
   ): Promise<T> {
     const maxRetries = 3
     let lastError: Error | null = null
@@ -158,7 +158,7 @@ export class SkillMcpManager {
   }
 
   // NOTE: tests spy on this exact method name via `spyOn(manager as any, 'getOrCreateClientWithRetry')`.
-  private async getOrCreateClientWithRetry(info: SkillMcpClientInfo, config: ClaudeCodeMcpServer): Promise<Client> {
+  private async getOrCreateClientWithRetry(info: SkillMcpClientInfo, config: ClaudeCodeMcpServer): Promise<McpClient> {
     const clientKey = this.getClientKey(info)
     return await getOrCreateClientWithRetryImpl({
       state: this.state,