I’m what situation is a min damage ring > full stay ring.

Is it when min damage on a weapon is low? Or when using forti min damage will be more beneficial etc

I understand fury use air but hat about conc Barb using legend sword?

simple answer: calculate your relevant stats (life, ar, def, avg dmg, ...) with both setups
if you want to go for a proper analysis, average 'number of hits to kill opponent' instead of 'damage output' (that requires information about your opponents setup as well ofc)

general answers to your question might be oversimplifications