While I take issue with the wording here, engagement reactions (responses and forces) occur whether the player chooses to engage, or the game forces engagement via engagement costs.
I think the following covers this issue...
Page 16 of the Core Rules:
Whether an enemy is engaged through an engagement
check, through a card effect, or through a player’s
choice, the end result is the same, with the enemy
and the player engaging one another. In all cases, the
player is considered to have engaged the enemy and the
enemy is considered to have engaged the player.