The core rule books (I've checked two) say to recover strain at the end of an encounter you roll a simple discipline or cool check and for each success, recover one strain.

In the beginner games (again, checked two) it says each PC recovers strain equal to their Presence or ranks in Cool skill. (whichever is greater) No rolling just straight recovery based on stats or ranks.

Is it either/or, or is one the actual right way to do it?

The rule from the Core Rulebooks is the "real" way to play it. The Beginner Games all simplify things for the sake of getting things moving for a bunch of first-timers. Of course, if you prefer the Beginner Game rule, it's your table, and you should keep using it if you wish. It just makes Discipline a bit less attractive as a skill option.

And don't forget the benefit of Second Wind as a Talent.