Neither

Usually (but not always, it depends on how the airline sells and issues the ticket) the point of departure dictates where the charge is made. So if you buy a trip starting in the US, it will be charged in USD, and it will not be a foreign charge. Doesn't matter if the airline is based in the US or not. If you buy a ticket that starts outside of the US, you'll likely get a foreign charge. At least that's how it works with e.g. BA, I'm not 100% sure if it works like that on all airlines as e.g. FI is happy to sell you a ticket starting in Europe and charge you in USD, can't tell if the price is actually just converted to USD but charged in the departing country or not.

For awards it usually does not work like that, e.g. AS awards (even those on partners starting abroad) will always be charged by AS in USD, and BA-issued awards will be charged in whatever country you live.

To be clear, even when the currency is USD, that doesn't necessarily mean the charge is not foreign, that depends on the point-of-sale. And the opposite, foreign currency charge, likely could also be with a US point-of-sale in some cases.

All of this is a lot easier when buying stuff at a physical store...