I don't understand well the difference between push-pull and half bridge. Someone told me it was totally different but as I understand it, half bridge is a generic name where we have 2 switches and a load connected in the middle. So I guessed that push-pull circuit is just a kind of half-bridge. Is that correct?
A half-bridge is a specific style of push-pull stage where the two active elements are in series across the supply and they alternately conduct to supply current to the load. The name usually refers to a switching output stage rather than one with linear control or only low power output.
The term push-pull refers to any type of circuit where there are two active devices that each drive one half-cycle of the signal - one pushes and the other pulls.
The devices can be in series across the supply or both be between the supply and ground and use a transformer to combine the two halves. They can be be used in linear mode for analog signals or just on/off for switching purposes.
The active devices can be any type of device such as FETs, electron tubes (valves), bipolar junction transistors etc.
Here are some examples:
Transformer coupled push-pull:
TTL output stage: