I’m not sure this is the right tagging, although the topic is a bit fraught. I think these lights are like the ones at Bolton-le-Sands (https://www.openstreetmap.org/node/10956035839). The crossing is _traffic_ light controlled, in that it’s basically only safe for pedestrians to cross when the traffic lights go red. But there are no _pedestrian_ lights.

So I think this should be crossing=traffic_signals (note ‘traffic’, not ’pedestrian’), but with button_operated=no and crossing:signals=shared.

Unfortunately the wiki doesn’t have an example specifically like this one. The nearest it has is the fifth example on https://wiki.openstreetmap.org/wiki/Key:crossing:signals#Examples

Let me know your thoughts! :)