Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.