> how can our meeting time have not changed if the exact instant it occurs has changed?

Because human time is not a numeric offset, but a name of relevant points.

> The choice in front of you is to pick a completely precise specification on which everyone agrees (the "instant of time") with an incomplete specification which you admit yourself may change in the future. My vote is to at least encode something unambiguous

The incomplete ambiguous specification is the one everyone agrees to, programmers just don't like them.