Good luck killing this Campsis radicans. Do not allow it to make long whips like this if you want to eliminate it. Tough aggressive vines.

I don't think it's a trumpet vine. The foilage is very similar but the shape of the individual leaves look different. That's what confuses me.

That is definitely Campsis. And the behavior you describe is classic - cutting it back to the ground or even digging up the main rootball does not guarantee its removal. It can regrow from root shoots located at considerable distance from the original plant and they spring up very rampantly and aggressively and can persist in doing this for years before the thing finally loses strength and gives up.

You can also put Roundup in a soda can, secure it to the vine so it doesn't tip over, cut the tip off a runner and stick it in the can. It will drink the Roundup and kill the roots and runners. Protect the soda can from getting water in it by putting a flower pot or bucket over it.