Is my understanding correct that
TA_UseTrial (using verified trials) will only need to access the LimeLM servers the first time it's called (namely to create the verified trial entry), and that after that it will always “just” check the trial data (as long as it's not tampered with of course) stored locally on the machine, without needing internet?
And, if not, when will it require an internet connection?
Yes, that’s correct. Also, internet will be needed when/if extending the trial.