Fix flaky test_run_no_log in DbApiHook

[henry3260]

Why

The previous implementation tried to work around this by using a fragile assertion: assert len(caplog.messages) in [1, 2]. However, this is an anti-pattern for two major reasons:

1. The previous implementation cannot clearly test what we want to test: By asserting the length of the logs rather than their content, the true intent of the test is obscured. It could theoretically pass even if the target message was never logged
2. It is also flaky: It still leads to CI failures if multiple background tasks happen to be garbage-collected simultaneously (making the log length 3 or more).

closes: #45774

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