Hello, I've just been re-reading the wiki and yes I think you might be right, although there is some extra nuance here.

Mapillary image here -
https://www.mapillary.com/app/?lat=54.05268786118398&lng=-2.7961026701102583&z=18.002058557365068&pKey=169029318452074&focus=photo

Interestingly the crossing is only marked on the exit from Parliament Street, and not the entrance (i.e. only across the bus lane). I think this may be because it is only this half that has a phase where the pedestrians can cross (to the island) while protected by lights.

There are (from memory) 3 phases at these lights.
1. Cable Street -> Parliament Street both directions, ahead-only, + pedestrian crossing Bulk Road
2. Cable Street -> Parliament Street (NE direction only) + Bulk Road -> Parliament Street left turn only.
3. Bulk Road -> all directions

All of these phases permit traffic to enter Parliament Street and conflict with pedestrians crossing - probably the reason for no markings on the NE-bound lane, and no signals.

Does that change how we tag it? Example 5 on the wiki certainly fits for the bus-lane. But the general traffic lane in the opposite direction is possibly just a standard unmarked crossing like you would get at any uncontrolled junction. However is that any different to example 5? Presumably there traffic can still turn into the junction and conflict with pedestrians - just have to give them priority?

As an aside, I think the labelling of Cable Street vs Parliament street is wrong - I think the area NE of the junction is all Parliament Street (you can see the sign on the mapillary image on the left hand wall under the green arrow), on the other side we can't use them as a source but MARIO and Google Maps both agree on a name while OSM has one side Cable Street and the other side Parliament Street!

My suggestion would be to tag as per your suggestion for the bus-lane side, but the other side I'm less sure. I don't object to apply your suggestions to both sides (possibly with different crossing:markings=*).

How does that sound?