I used an old credit card to carefully “cut” the battery from the base. You have to exercise some patience here, because the battery is very thin and bendable and probably dangerous to bend too much. The thin, tape-like adhesive stuck to the case as the battery itself came off, so the new battery stuck just fine to the old adhesive.
These instructions are not for the Mid 2012 A1278, but for a much older MacBook Pro. Pretty confusing.