In Nederland, an official zebra crossing includes a traffic sign L2, and the stripes can have prescribed size limits and colours. Colours? Yes, e.g. the zebra pattern can be brownish/reddish and whitish/greyish paving stones. But the pedestrian priority does not require the L2 sign, and if the sizes and colours do not fully meet the requirements exactly it’s still a pedestrian crossing with pedestrian priority.
So, crossing:markings is the tag to set; crossing_ref=zebra only adds that it meets the official requirements posed to the operator.
Sidebar:
Personally, I do not intend to check colours, size and signs for all crossings. We have a Dutch JOSM preset that sets the tag though, so I am sure that many mappers do set crossing_ref=zebra without checking. We will probably do a cleanup some time in the future, setting crossing_ref=zebra only when there is an L2 sign and removing it for all other zebra marked crossings. (We have an official source for the signage).
In the meantime, I have started to add crossing:markings=zebra to all zebra crossings, using a data source which used aerial imagery to detect zebra patterns. Lots of false positives and false negatives; still, I estimate that over 95% is actually a zebra-patterned crossing, and most of the false positives are recently removed or shifted zebra crossings, which can be verified by more recent imagery.
I am not yet removing crossing=zebra set by other mappers.