I write down the time when I switch tasks, and also I try not to switch between many different tasks/issues. As for the details, the app we use for time tracking has a calendar input interface which is precise upto 5 mins, with all our gitlab issues synced as billable tasks, so that is pretty straightforward. Also, I tend to leave a small note on what part I worked on so that when looking back the time is justified. If you don't have that maybe drawing bars on google calendar in a separated "billing" calendar could help.

Keep in mind, that when it comes to being very punctual on time you can play the politics back (e.g. when managers hammer on clocking in on time, you get to clock out on time too) and claim budget for work items that took shorter than expected.

I found Timely https://memory.ai/timely useful for my case.