There is ofcouse a difference between the usage of a halberd when fighting in a formation and when fighting solo with enough space. The videos from you, Fidelis, primarily seem to show the poleaxe in action - while sizes varied, the halberds were usally (quite) a bit longer than poleaxes, so using them as reference probably has it's limits. (To be fair, it's often not easy to say what is poleaxe and what is a halberd. The same thing applies to other weapons as well - especially swords. When is a one-handed sword a bastard sword? When is a bastard sword a greatsword? The aim to categorise historical weapons is more a modern idea.)
From what I have read and gathered, the halberds would be used for thrusting attacks most of the time - if the target was missed, intentionally or not, the halberd head would then be behind the opponent and could be pulled back, inflicting damage with the hook/thorn. Doing this could also throw your opponent off balance (especially by attacking the legs), or could be used to pull riders from horses. Ofcouse a strike in an arc - to penetrate armour with the spike/thorn - was also possible and common, but only if the situation allowed it. The axe blade itself seemed to be there more for cutting during a thrusting attack and for the hooks, and not so much for hitting people with it - but that probably depends alot on the length and the design of the halberd and the situation.
While fighting in a unit/formation, the soldiers would mainly use thrusting attacks. Soldiers further back could probably not use the "pull-back attack", as to not endanger the ones in the front. A strike with the axe blade would presumably be quite limited (due to space), but I guess some would definitely try to hit the opponent (on the head) from above, even though you can't deliver a strong blow.
Edit: After writing this, I found this video on youtube - I think the guy makes some very good points:
https://www.youtube.com/watch?v=GsckeyktMS0
