Hi. Sorry for the delay in replying.

The following is my take on what is a crossing object (node) and what isn't a crossing object (way).

With the node/way for crossings. it's the highway tag that defines what the object is (similar to a noun). The subtags just add detail about the object (similar to adjectives).

Node: 
A crossing on a road is mapped and definied as an intersection NODE on the road. Without a "highway=crossing" tag this is an undefined node. Even if a footway is connected to the node, it is this "highway=crossing" node that is further defined according to what type of crossing it is. with tags such as "crossing=traffic_signals". 

Way: 
A "footway=crossing" is added to "highway=footway" to indicate what type of a FOOTWAY it is. Without a "highway=footway" tag a "footway=crossing" is an undefined way. A "footway=crossing" is definied as the width of the roadway from one kerb to the other kerb, i.e. it's part of a roadway rather than being a true footway,

This 'crosswalk' FOOTWAY never has tactile paving or traffic signals - these are always on the sidewalk/crossing island. Tagging with "crossing:island" on the footway is just as meaningless as it would be if tagged on the road - this tag was created for crossing nodes to avoid use of "crossing=island".

The 'crosswalk' FOOTWAY can be tagged with surface (a tag not added to a crossing node!) and this way is either marked/zebra or not marked.

To avoid confusion, I think "footway=crosswalk"would be a better way to subtag these footways. The iD editor is particularly poor at distinguishing between the nodes + ways, and I'm sure lots of users are being led astray by iD.

In the last few months, with their preset "marked crossing" for nodes, iD have at least stopped mistagging these nodes as "crossing=marked" and now use the correct "crossing=uncontrolled".

If you look at he last 2 tagging examples at https://wiki.openstreetmap.org/wiki/Tag:highway%3Dtraffic_signals#How_to_map_(new), these do not add "crossing=traffic_signals" to the crosswalks ('Green ways').

I can see that at https://wiki.openstreetmap.org/wiki/Tag:footway%3Dcrossing (the English version) User:Victor.yarema added (in Feb 2021) an example with the "crossing=traffic_signals" subtag.

Having looked at the other 9 main language versions: this "crossing=traffic_signals" did get added into the German wiki (in Nov 2022), but it is not present on the French, Spanish, Italian, Portuguese or Polish etc wikis. As such, it appears to not be an approved change to the wiki.