The SEP says that moral realism means thinking that (some) morality exists as objective fact

"Morality exists" and "as objective fact" are interpolations. The SEP article just defines moral realism as the claim that at least one moral statement is true (in the correspondence-theory sense of 'true'). So moral realism is success theory (as contrasted with error theory), or success theory + moral-correspondence-theory.

some other process which would lead all right-thinking minds to agree about it

'Right-thinking' in what sense? Whence in the SEP article are you getting this claim?

'The SEP says' is also a mistake. The article you linked to defines 'moral realism' one way; the article on moral anti-realism defines it in a completely different way. (One that does try to make sense of an 'objectivity' constraint.) Good evidence that this is a bad word.