Fixes gh-159303.

With no custom handler set, call_exception_handler() called default_exception_handler() in the caller's context. gh-96704 made a custom handler run in the task's or handle's context, and its PR left the default handler for later. This moves the lookup into a helper and uses it for both.

Both handlers run in ctx.copy(). Without the copy, reporting from inside the task itself ('task': asyncio.current_task()) raises RuntimeError: cannot enter context, and a custom handler is silently skipped today.

Three tests in test_base_events:

• the first fails on main (the default handler sees the caller's context);
• the third fails on main (the custom handler is skipped);
• the second passes on main and fails only if the copy is dropped.

The docs note both changes. The issue's repro now prints req-1 twice. ./python -m test --fast-ci -j8 test_asyncio passes.

Prepared with AI assistance (Claude), then reviewed and tested before submission.