Why bother? I use the expression 'tax disc' often enough. Indeed, when I'm queueing at the Post Office (Yes! I prefer to do it the old-fashioned way), I don't start thinking, 'I'm getting my VED disc'. When I reach the counter I ask for a 'tax disc'. When the first of the month comes round and I dutifully stick it on the windscreen, I'm thinking of it as a 'tax disc'. Nothing else.
I admit, sometimes I may be guilty of a bit of pedantry - on this forum too! May I redress things by insisting, it really doesn't matter!
What's far more meaningful to the motorist maybe, is to retort: "I pay for a tax disc too, although my car's parked on the drive out of everyone's way". I appreciate that not every cyclist has that up their sleeve.