We can argue all day about the technicalities of whether you steer or counter steer and at which speed.
I think the relevant points of the OP have been answered in this thread:
1. Push left go left, push right go right.
2. Actively pushing to steer is many times more effective than just leaning (and unconsciously pushing in the correct direction anyway), so knowing to do this correctly when stressed might save your ass in an oh-shit situation.