There’s currently no way to tag who has priority/right of way on a crossing, which is probably the most interesting tag.
agreed, I think mappers and people are typically assuming this is implicit. On top of the legal situation, there are also cultural habits. E.g. in Italy, pedestrians have the right of way on zebra crossings, but in the south of the country, motorists are typically expecting pedestrians to let them through nonetheless, and while you might have the legal right to walk first, it may be better advice to wait a second rather than going to the hospital for some weeks because you insisted on your right.
I think the situation is similar in other countries as well.
For practical usefulness, we would have to record what is the actual situation and not just the theoretical legal situation.