Yes, and the little stove that slips over the cup also.

If the cup "sticks" to the canteen, sometimes it is possible to gently bend the cup a bit to free things up.  Easy does it, and having the canteen a little tight to the cup ––and not rattling–– is better than too loose.

Recall that the canteen, and cup are more-or-less kidney-shaped, as seen from above.  The cup most often becomes squeezed in at the short  "diameter" of the kidney shape.  The easiest thing to do is get a couple of  fairly narrow wooden boards, and pry the boards against each other to open up the top a bit.  The boards, when being used, will resemble a "V", with the bottoms touching in the bottom of the Cup, and the open end of the "V" up out of the cup, being grasped by your hands.  Easy does it.

If your cup has been dented, you will need the point of an anvil to act as a backstop, and you will tap around the edges of the dent, letting the anvil push the dent out.  Some dents are tougher than others to repair, and at some point it is easier to get a new cup.  An auto body repair guy, or maybe a blacksmith or sheet metal guy might be able to repair a badly dented cup, but their fee for doing so might exceed the cost of a new cup.

I dunno; one might have to experiment with possibly enlarging the feed opening and/or the vent holes to optimize it for twigs and so forth..