Internal premolded heel counter fit to the last.
Over on the Soccerpulse forums there has been some questions about internal and external heel counters used in soccer shoes (and also other athletic shoes). here is some more info and pics.
More pics and info after the Read More link below.
1. all shoes do have a heel counter or counter reinforcement. The good, hard ones, us a plastic, TR (thermoplastic resin) or nylon material premolded to fit the heel of last. As you point out not all reinforcements are equal however. The onesyou mention that can "push in" have a cheaper and less stiff reinforcement- usually a type of "chemisheet". This is like a paper with a chemical in it that hardens when activated bny either heat or another chemical.
The important thing about a heel coutner is that it fits the heel of the last. The purpose of heel counter is to provide support and ensure good fit.
2. An external heel counter as Romeo mentioned does the same thing, but is on the outside of the heel. Usually it is a TPU material.
Internal heel counter
External heel counter