An enemy has an ability that states Forced: When it engage a player it has +1 attack, or does 5 damage to a hero etc

Is this ability triggered also when a player chooses and engages an enemy from the staging area?

Or the enemy has to engage you only through the engagement check in order to trigger the ability?

Any time an enemy moves from the state "not engaged with a player" to the state "engaged with a player", it has engaged the player and the player has engaged it. The reason it moved between those states does not matter.