Definitions of a "backpack" are quite vague....I mean I have hiked for two weeks in the high Sierras with a "backpack" with everything I needed...food, tent, sleeping bag, clothes, etc. And it DEFINITELY would not fit under a airline seat.
Your mention of a "drawstring" backpack (I would call that a small "daypack") is better. Yes, those, and other small day packs, fit under the seat...we travel with one frequently. But nothing bigger will fit under the seat.