I think perhaps you are being misled by the title which the ID editor applies to crossing=traffic_signals nodes. It does title them as “Crossing With Pedestrian Signals”, but if you look lower down, the Type is “Crossing With Traffic Signals”. I think it’s a bug for ID to title them “Crossing With Pedestrian Signals”.

In any case, the wiki gives the canonical meaning of the tags, not whatever labels ID applies to them. See https://wiki.openstreetmap.org/wiki/Tag:crossing%3Dtraffic_signals