This is a very skewed test I think. While your result and intuition are likely to be true, the test does not prove it. The streetlights test measures the response to a sudden change with no previous indication. While playing an FPS, you get multiple sources of information which converge before a specific event.
To be more precise, you know what to expect and when - even without counting exact seconds, if you see someone running, then disappear behind something - you know when to expect him to enter your view again. Then you've got sound which additionally helps you in orientation. Then you've got the time it takes between you spotting something and getting the correct aim, while you see the target moving - you're aligning your aim with the moving object and deciding when they meet, rather than just measuring a response to an impulse.
To be more precise, you know what to expect and when - even without counting exact seconds, if you see someone running, then disappear behind something - you know when to expect him to enter your view again. Then you've got sound which additionally helps you in orientation. Then you've got the time it takes between you spotting something and getting the correct aim, while you see the target moving - you're aligning your aim with the moving object and deciding when they meet, rather than just measuring a response to an impulse.