Is sales tax determined by the state in which the SELLER is located, or the state in which the PURCHASER is located?

Online sales have been a commonplace part of life for 30 years, and my understanding over that entire period has always been that sales tax is only collected if the seller maintains a presence in the same state as the purchaser; otherwise, if the seller and the purchaser are in different states, no sales tax is collected (although the purchaser may be nevertheless obliged to pay a use tax to the state).

I (in Michigan) am trying to order something online from a merchant in Arizona who insists that they are required to charge me sales tax at their local rate. Can someone confirm whether or not I'm correct in thinking that they are wrong to do so?

my understanding over that entire period has always been that sales tax is only collected if the seller maintains a presence in the same state as the purchaser

This understanding is incorrect. This U.S. Supreme Court imposed rule was overturned by the U.S. Supreme Court in the case of South Dakota v. Wayfair 585 U.S. 162 (2018). In South Dakota v. Wayfair, the U.S. Supreme Court held that:

Because the physical presence rule of Quill is unsound and incorrect, Quill Corp. v. North Dakota, 504 U. S. 298 [1992], and National Bellas Hess, Inc. v. Department of Revenue of Ill., 386 U. S. 753 [1967], are overruled.

States now can and do impose sales taxes on sales with a destination in their state from a seller with no physical presence in the state.

Note also that the answer from bdb484 is not incorrect. Whether a state or local government actually does actually impose a sales tax where the buyer takes delivery, as opposed to where the seller is located, is a matter of state law.

Furthermore, even when sales taxes on remote sellers were unconstitutional prior to 2018, a state could still impose a "use tax" directly on the buyer that applied to purchases from remote sellers, but the seller didn't have to collect it. Use taxes were invented to circumvent the constitutional limitations of Quill and Bella Hess where those cases were good law.

It depends.

Sales tax regimes can be set up for either origin-based sourcing or destination-based sourcing. To simplify a bit, origin-based systems assess tax based on the location of the sale, while destination-based systems assess tax based on the location where the buyer will take possession of the goods.

Like most states, Michigan uses a destination-based system, but Arizona is an origin-based state. It therefore charges tax at the rate of the seller, regardless of the location of the buyer.