Some thoughts on this:

crossing=zebra — I wouldn’t use this tag at all. If you don’t know if there are traffic signals or not, crossing=marked + crossing:markings=zebra can be used instead. crossing=marked has about 2 442 000 uses worldwide on taginfo, and crossing=zebra has about 464 000 uses. Also I believe crossing=zebra is deprecated, and as you said its meaning is confusing and it has been widely misused.

highway=crossing + kerb tag — if the kerbs are not micro-mapped with barrier=kerb nodes this is okay, but if they are mapped separately, according to the OSM wiki the kerb tag should only be on the barrier=kerb nodes.

“… but preferably tag the details on either the node OR the way, not on both at the same crossing” — I’d add tags like crossing, crossing:markings, crossing:signals to the crossing node, because this is an older and better supported tagging scheme. Tagging only the way with crossing tags is not well supported now. Though tags like bicycle=designated, segregated should go on the way, like you said.