If we were to use something like designation=GB:zebra (or a country code prefix in some other tag) that would make it absolutely clear that we’re only referring to one where UK rules about pedestrian precedence and road markings apply.
We also have a lot of not-zebra crossings on private land like car parks and industrial estates in the UK. They have the zebra stripes, but no legal enforceable rules, give way markings, Belisha beacons, restricted area with zigzags, etc. I usually tag these with crossing=uncontrolled + crossing:markings=zebra + not:crossing_ref=zebra, which is a very unsatisfactory way of doing it.