Who is weakening the conditions for a theory being "consequentialism"? The thing described by Peterson seems perfectly in line with consequentialism. And his point about asymmetry among moral theories remains.
Well, there are a lot of things that get called "consequentialism" (take a look at the SEP article for a similar point). I personally find that "consequentialism" connotes to "agent-neutral" in my head, but that may just be me. I feel like requiring neutrality is a more interesting position precisely because bare consequentialism is so weak: it's not really surprising that almost everything is a form of it.
There's also the possibility of accidental equivocation, since people use "consequentialism" to stand for so man...
This was demonstrated, in a certain limited way, in Peterson (2009). See also Lowry & Peterson (2011).
The Peterson result provides an "asymmetry argument" in favor of consequentialism:
Another argument in favor of consequentialism has to do with the causes of different types of moral judgments: see Are Deontological Moral Judgments Rationalizations?
Update: see Carl's criticism.