I don't think the infrastructure is designed for a cyclist to turn right onto the cycle path from the cycle lane. It assumes that cyclists are already on the cycle path and crossing the road, or are on the cycle lane and progressing along the road. I can't see anything that says give way to right turning traffic for the van, or as mentioned, for the cyclist.
In fact, giving priority to right turning traffic, including traffic turning across a cycle lane, and that close to another junction, is such a fantastically dangerous thing to do that I would be astonished if that was the intent, and I would expect it to be clarified by a sign saying 'give way to right turning traffic' or something.
I can't interpret that junction any other way than 'give way to traffic crossing on the cycle path'. The fact that 'everybody gets it wrong' and that there appear to be quite a few experienced cyclists here disagreeing with the OP's interpretation is quite telling. OP, you mentioned the highway code upthread, perhaps you could pooint us in the direction of the specific sections of the code that deal with a junction like this one, or which you think are relevant?