fix: drain stdio responses after stdin EOF

[fengjikui]

Fixes #2678.

Summary

This keeps the stdio server write side alive briefly after stdin EOF so responses for already-accepted requests can flush before the process exits.

The default server/session behavior is unchanged for other transports: read-stream closure still closes the write stream and cancels in-flight handlers. Only MCPServer.run_stdio_async() opts into the bounded drain path.

Root Cause

For file-redirected stdio input, the server can read and accept tools/call requests, then immediately hit stdin EOF while those handlers are still mid-await.

Before this change:

• BaseSession._receive_loop() wrapped the read and write streams in one async with, so read EOF closed the write stream.
• Server.run() then unconditionally cancelled the handler task group.
• Accepted tool calls could run far enough to be visible in server logs, but never send their JSON-RPC responses to stdout.

Change

• Add an internal session option to decouple write-stream closure from read EOF.
• Add an internal Server.run() drain option that waits for in-flight requests to complete, bounded by a timeout, then falls back to cancellation.
• Enable that drain only for the stdio server path.
• Add regression coverage for both successful stdio drain and drain-timeout cancellation.

Validation

• Reproduced FastMCP/stdio: in-flight tool responses dropped on stdin EOF when input is bash-redirected from a file #2678 before the fix: stdout had only id=0, missing id=1 and id=2 despite both CallToolRequest handlers being entered.
• Re-ran the same repro after the fix: stdout has id=[0, 1, 2] and both tool response texts.
• uv run --frozen ruff format src/mcp/shared/session.py src/mcp/server/session.py src/mcp/server/lowlevel/server.py src/mcp/server/mcpserver/server.py tests/server/test_stdio.py tests/server/test_cancel_handling.py
• uv run --frozen ruff check src/mcp/shared/session.py src/mcp/server/session.py src/mcp/server/lowlevel/server.py src/mcp/server/mcpserver/server.py tests/server/test_stdio.py tests/server/test_cancel_handling.py
• uv run --frozen pytest tests/server/test_stdio.py tests/server/test_cancel_handling.py tests/server/test_lowlevel_exception_handling.py -q -> 13 passed
• uv run --frozen pyright src/mcp/shared/session.py src/mcp/server/session.py src/mcp/server/lowlevel/server.py src/mcp/server/mcpserver/server.py tests/server/test_stdio.py tests/server/test_cancel_handling.py -> 0 errors
• UV_FROZEN=1 uv run --frozen strict-no-cover -> passed

Note: I also attempted ./scripts/test; local HTTP/SSE/WebSocket tests were affected by my machine's proxy environment (socks5://127.0.0.1:7890), so I did not count that as a clean validation signal.

AI Assistance Disclosure

I used AI assistance to help trace the shutdown path and draft the patch. I reviewed the changed code and ran the validation above locally.