Here in the US it can be a debated topic. Texas to my understanding, Northern Mexico as it were, is the birthplace of chili. I'm in California and we tend to reimagine most every dish so I wont give my answer. Beans may or may not have been part of the original recipe. Amongst chili aficionados you might not get a clear answer .
I'd say use whatever chili's you have tho. Not sure of this articles accuracy but thought it was worth a read. https://whatscookingamerica.net/History/Chili/ChiliHistory.htm
Good question.... With all the variations, I would think it's another one of those things that have a half-dozen criteria, and you need to meet all but one for it to be considered Chili.... give or take.- it would have to be spiced with hot chilis (ground/powdered or fresh), and it would have to contain cumin- it would have to be the consistency of a stew - it would have to have small bits of whatever is in there.... the size of a chickpea or smaller- beans - meat/poultry - tomato as the baseIn that case... chili with or without beans, white bean chili, veg chili... all qualify. It's similar to the question of: when is a SAK no longer a SAK.