No, this is wrong. It is quite simple, really - same diameter (more precisely, entrance pupil), same light.
You are mistaking light candelas/luxes with lumens.
No, you are.
You get the same amount of light in lumens, but not the same light density (luxes/candelas), since the light is spread all over the sensor.
Right, but you are the only one insisting that intensity has anything to do with the discussion, so ... you get the idea.
But anyway, there's no point discussing what's already discussed elsewhere, so please better read here:
Yes, good reading. I have been referring to it for years. Did you get to the "total light" part of it?