What criteria does the empeg use to update the 'plays' counter? Is it a fixed amount of time that a track has to play for, or a percentage of the track length or something else?

IIRC the track has to play for 5 seconds before it is counted. This initially created a bug for tracks which weren't even that long, but this bug has been ironed out with the 2.0 final release.

Hopefully in 3.0 they'll use percentages instead for plays/skipped counter updates.

What criteria does the empeg use to update the 'plays' counter?

As was already said, the track has to play for 5 seconds. But there is one other criteria that must be met: The track has to be exited before the count will get written to the scratch partition. Either by the song finishing normally, or by you pressing the button to skip to another track.

So if the song plays for five seconds and then you yank the power while the song is still playing, it won't show the updated count.

At least that's what I recall one of the Cambridge folks telling us once.

Also it's my impression (and correct me if I'm wrong) that restarting a track does not increase the play count either.

Right. I think that's the reason the "gotta exit the track" limitation is there in the first place.

What if I fast forward? Would that be 5 seconds of fast forwarding time or 5 seconds in normal time and thus 10-30 seconds of actual track time?

5 seconds of real track time. Dunno if it skips the count if you FF past the 5-second mark.

You'll have to pass the 5 seconds playing time mark. How you do that, either by regular playing of FF doesn't matter.

I do this all the time, but for me this is much easier to do with my Kenwood RC-600 remote because it has separate FF & REV buttons. (the "+" & "-" buttons on the photo - I love this remote!).
I don't think I would bother with the original remote because a long button press (which is needed to FF) takes about 5 seconds to complete anyway so I think I would just let it play then.