re: we know that price determines demand...
Price alone does NOT determine demand. Price obviously has an impact but it is not sufficient to explain varying demand for like goods. The dark art of setting the correct price for a good is a a key part of bringing a good to market successfully. This price may well be set higher compared to a like good, precisely in order to increase demand for the higher priced good vs the other. Cases in point: detergents, toothpaste, trainers, computers, USB cables, etc etc.
My point being that one of the issues with economics is the insistence on vastly over simplified theoretical models, whether presumptions about human nature or the mechanics of supply and demand (such as price = demand) which far too often fail to match real world experience.