Eureka!
After further experimentation, I think we've cracked it. Being fairly certain that there were no differences between IE10 on my computer and IE10 on my husband's computer, we started to consider what differences there might be between our computers that could be triggering this issue, and one obvious difference was our monitors: I have a rather old 4:3 format monitor (1024 x 768 pixels), whereas his is a newer 16:9 one (1366 x 768).
Resetting his monitor resolution to 1024 x 768, we could then replicate the issue ("Save image" rather than "Save to your computer") on his computer, in both IE10 and Firefox. Reverting to the default resolution took him back to the correct version of the menu.
One slight spanner in the works with this is that we have an ancient laptop running XP and IE8, also with a display resolution of 1024 x 768, and we're getting the correct "Save to your computer" option on that. However, the html code for the page has some comments at the top which I think suggest that it uses different javascript files depending on whether you have IE7, IE8, IE9 or an unspecified version later than IE9.
So presumably having IE10 is not of itself the issue, but the update nevertheless triggered it, and I'm guessing that the different javascript file used by IE10 and later contains something about screen resolution.
And this takes me back to one of my original thoughts, that somehow something in the page's code was identifying my computer as a tablet rather than a desktop. Well, if you assume that only tablets will have a screen resolution at 1024 x 768 or less these days, then maybe there was something in that thought after all - and I did suggest this to Ancestry nearly a fortnight ago.
So now I need to report this back to Ancestry and see what they make of it. But I'd be interested to hear anyone else's further thoughts on this.