Why does the test wait a random amount of time before turning green?
Because the wait varies between 1500 and 4500 milliseconds precisely to stop you from clicking by rhythm without actually watching the color change. A fixed delay would let you memorize the moment and click ahead of time, invalidating the measurement.
Why does clicking too early not count as a result?
Because the phase switches to "too soon" instead of recording a near-zero time, and the same round is redrawn from scratch. That stops a lucky or impatient click from artificially dragging your average down.
Is the time measured in the browser the same as pure neural reaction time?
No, the number includes screen latency up to a full frame of delay at the refresh rate, input device latency, and the time the browser takes to dispatch the click event. The result measures "stimulus to registered event", always somewhat larger than the purely neural reaction.
How does the screen refresh rate influence the measured time?
At 60 Hz, each frame takes about 16.7 milliseconds; the switch to green can be ready in code and still wait until the next frame is drawn to actually appear on screen, adding up to that amount to the total measured time.