Implement MCP spec-compliant stdio shutdown sequence

Following the MCP spec recommendations, the shutdown sequence now:
1. Closes stdin first to allow graceful server exit
2. Waits 2 seconds for the server to exit on its own
3. Only escalates to SIGTERM if the server doesn't exit
4. Finally uses SIGKILL as a last resort

This unified approach works consistently across all platforms and gives
well-behaved servers a chance to exit cleanly without signals.

Co-Authored-By: davenpi <noreply@github.com>

Author: felixweinberger

src/mcp/client/stdio/__init__.py

@@ -177,17 +177,38 @@ async def stdin_writer():
         try:
             yield read_stream, write_stream
         finally:
-            # Clean up process to prevent any dangling orphaned processes
+            # MCP spec: stdio shutdown sequence
+            # 1. Close input stream to server
+            # 2. Wait for server to exit, or send SIGTERM if it doesn't exit in time
+            # 3. Send SIGKILL if still not exited
+            if process.stdin:
+                try:
+                    await process.stdin.aclose()
+                except Exception:
+                    # stdin might already be closed, which is fine
+                    pass
+            
             try:
-                process.terminate()
+                # Give the process time to exit gracefully after stdin closes
                 with anyio.fail_after(2.0):
                     await process.wait()
             except TimeoutError:
-                # If process doesn't terminate in time, force kill it
-                process.kill()
+                # Process didn't exit from stdin closure, escalate to SIGTERM
+                try:
+                    process.terminate()
+                    with anyio.fail_after(2.0):
+                        await process.wait()
+                except TimeoutError:
+                    # Process didn't respond to SIGTERM, force kill it
+                    process.kill()
+                    await process.wait()
+                except ProcessLookupError:
+                    # Process already exited, which is fine
+                    pass
             except ProcessLookupError:
                 # Process already exited, which is fine
                 pass
+            
             await read_stream.aclose()
             await write_stream.aclose()
             await read_stream_writer.aclose()