I'm not sure what SSI teaches now, with PADI I learned to donate the octo.
With GUE I learned to donate the primary.
Their reasoning is that when someone is out of air you should give them one that you're 100% sure is working as they're probably running out of breath.
I've seen too many divers drag their octos through the sand/sediment to not agree with them. If I'm out of air I'd rather not get a mouth full of mud first. Besides that, if someone is starting to panic they're likely to take the primary from you anyway because it definitely works and is the easiest to find.
This is also the reason that agencies that focus more on tec diving prescribe having the primary on a long hose for easy sharing and the secondary on a shorter hose hanging on a necklace for easy access. That way you're also sure it's not dragging through the mud.