False positives mean you throw away ham as spam (terrible), false negatives mean you let some spam into your mailbox (not a big deal). But false positives also mean you diagnose benign as malignant (annoying), and false negatives also mean you diagnose malignant as benign (fatal).

So depending on what you are looking for, it varies whether false negatives or false positives are worse. Or maybe they can be equally bad.

I don't think you need to rename positive and negative as long as positive is understood as unusual and negative is understood as normal.