Charles, I did read your comment: when I quit Word overnight then the next day it dates correctly; if I simply close it, it doesn't, until the afternoon.
That's Word 2016 on the Mac Mini. The problem with quitting Word is that it can take 5 minutes to re-open, whereas it opens immediately from closed.

However, Word on my Windows 10 dates correctly, regardless of morning or afternoon, whether I simply close it (my normal action) or quit it (fairly rarely).

As you said, "Word for the Mac is usually, these days, behind Word for Windows. (I remember when Word for the Mac was ahead of Word for the PC, especially before Windows.)" I too remember those days, hence disappointment.