For games in general, an "engine" is "do more." You increase how much you can do with your turns (yes or how many turns you get). When engine games started to be a thing, they were also called "snowball" games, after snowballs getting bigger as they roll down hills. Your engine snowballs; it does more and more.
In Dominion there are three basic ways you can do more. By default you get 5 cards a turn, can play one Action, and can gain one card. You do more by drawing more cards, playing more Actions (not necessarily terminals), and gaining more cards. Gaining more cards includes having the economy to pay for them (or using Workshops, which provide their own economy). So getting better economy is part of this too.
When people talk about engines in Dominion - and of course all that's useful is what people use the term to mean, not what any one person would like it to mean - they tend to mean decks that include all three of these elements. You have to be drawing more cards, playing more than one Action, and gaining more than one card.
I don't think there are people who seriously talk about a deck that has no draw or can only play one Action a turn as an engine. If you can only gain one card, well that can be good, drawing your deck with Labs say, and the Labs do snowball, to a point. But it's limited, even if the Labs never run out. To some people it may count but I think to most it does not; it's "drawing your deck" not "an engine."
It's all a loop feeding itself. Drawing, playing, and gaining more cards all help you draw, play, and gain more cards.
A weak engine is still an "engine." You would like to draw your whole deck every turn, play everything, and gain lots of cards; but just getting part of the way there is still an engine.