astrophotography – How to accurately mark the digital SLR plan?

I want to know the exact UTC time of when the trigger is pressed. Precision in the order of milliseconds would be ideal, but a value greater than 1 second is acceptable.

The details of the properties of the photo are not precise enough. The indicated time is at best in UTC minutes.

Similarly, is there a lag between clicking the shutter and opening the shutter? Yes, of course, but how much? and does it depend on the time of exposure and something else?

I shoot satellite passages in the night sky, with a final goal of determining the orbit. Precision matters so much!