Does voltage limit current?
So I thought, that the resistance might affect this effect?


Absolutely.

The resistance of a 'short-circuit', including the contact resistance from the supply terminals to the wire, is not zero, but finite. 

If the supply is being asked to deliver (say) 1v to a 0.1ohm resistance, then ideally it would take 10A. If the supply limits at 5A, then the output voltage would fall to 0.5v so that's all it's delivering. Reducing the requested voltage to (say) 0.2v would result in only 2A flowing, which it would deliver happily.

Congrats on knowing the difference between effect and affect.