Is this the same thing for the purpose of triggering "engagement" related card effects?

The rules state: Note that any time an engagement occurs (either by player option, by an engagement check, or by a card ability), the end result is the same: the player is considered to have engaged the enemy and the enemy is considered to have engaged the player.

So a card like the Elder Nameless Thing triggers its forced effect both when you optionally engage it and when it engages you via an engagement check?

Yes, no matter the way the enemy engages you, its effect will trigger.

Thanks for that. I was taking the wording too literally.