20×2.125″ tyre currently fitted on rim – will 20×1.95″ tyres fit on the same rim?
What governs width across the bead at the rim?

Yes, both will fit the same rim.

The important number is the ETRTO number, usually on the side of the tyre. For those tyres it’ll be xx-406, so they’ll fit any 406 rim. There is another size also called 20″, but it’s 451 so almost 2″ bigger.