How do they check this kind of thing? Do they ask the buyer to display the Amazon page on their smartphone if it's just a small discount? Or do they always check on their own devices? (Amazon reportedly sometimes displays different prices for different accounts.)

In my experience, if you need a retail clerk to do something for you and the first person turns you down, you can often just go to a different clerk (if possible in a different shop) and try your luck again. (I've done this 1) to get my phone company to change my phone number despite not quite fulfilling the requirements 2) to get an MP3 file of a store's theme song.)

Now I believe trying to fake an Amazon page in order to get a price-matching discount is most likely fraud, but it still seems like some people who aren't fraudsters in general (e.g. teenagers) would be trying this kind of thing.

As of the last time I heard their policy (within a couple of years, from floor/cash register staff), Fulfilled By Amazon doesn't cut it -- they will price match only if Amazon is the vendor.