fix(examples/client-multi-server): align with upstream client/stdio subpath split

Upstream main split StdioClientTransport into the './stdio' subpath of
@modelcontextprotocol/client to keep the root entry free of process-spawning
runtime dependencies (child_process, cross-spawn). Two changes were needed
to align this example with the new layout:

1. src/index.ts: import StdioClientTransport from
   '@modelcontextprotocol/client/stdio' instead of from the root entry.

2. tsconfig.json: add path mappings for the new client/stdio subpath and
   the core/public subpath, matching the client-quickstart example. Without
   these mappings, the example's typecheck resolved client/stdio through
   the package's exports map to dist, which produced a race condition with
   parallel build steps that clear dist before rebuilding.

Local typecheck:all, lint:all, and build:all all pass.

Co-Authored-By: Tadao <tadao@travisfixes.com>

Author: travisbreaks

examples/client-multi-server/src/index.ts

@@ -11,7 +11,8 @@
 import { resolve, dirname } from 'node:path';
 import { fileURLToPath } from 'node:url';
 
-import { Client, StdioClientTransport } from '@modelcontextprotocol/client';
+import { Client } from '@modelcontextprotocol/client';
+import { StdioClientTransport } from '@modelcontextprotocol/client/stdio';
 
 const __dirname = dirname(fileURLToPath(import.meta.url));
 

examples/client-multi-server/tsconfig.json

@@ -12,9 +12,13 @@
         "forceConsistentCasingInFileNames": true,
         "paths": {
             "@modelcontextprotocol/client": ["./node_modules/@modelcontextprotocol/client/src/index.ts"],
+            "@modelcontextprotocol/client/stdio": ["./node_modules/@modelcontextprotocol/client/src/stdio.ts"],
             "@modelcontextprotocol/client/_shims": ["./node_modules/@modelcontextprotocol/client/src/shimsNode.ts"],
             "@modelcontextprotocol/core": [
                 "./node_modules/@modelcontextprotocol/client/node_modules/@modelcontextprotocol/core/src/index.ts"
+            ],
+            "@modelcontextprotocol/core/public": [
+                "./node_modules/@modelcontextprotocol/client/node_modules/@modelcontextprotocol/core/src/exports/public/index.ts"
             ]
         }
     },