There are a number of rt-PCR (reverse transcriptase polymerase chain reaction) tests that detect the RNA from the SAR-CoV2 virus (Covid 19). These tests take anywhere from 45 minutes to several hours to run depending on the extraction technique and generally require a full clinical pathology lab setup (think a hospital or LabCorp facility) and are reported out by the site that process them, and they are usually run in batches. The "rapid" 15 minute or less tests are typically antigen tests that test for surface proteins of the virus(such as Abbott BINAX) or the "molecular" tests that react to parts of the virus' nuclear material (such as Abbot ID now) These can be done and read onsite at an airport, pharmacy, or doctor's office. The latter tests are not universally accepted for international travel or travel to Hawaii. Interestingly, the US CDC regulations do permit the use of all of the types of test for re-entry to the country by citizens- but some airlines insist on rt-PCR regardless.
rt-PCR is the gold standard of testing for covid with a very high accuracy. The rapid tests, not so much.
Having been involved with collecting and processing of several hundred thousand of these things over the last 14 months, I bet the first Walgreen's one had the Abbott ID-now molecular test which is often confused with a "real" rt-PCR which was only available at the second store.
Probably more than you wanted to know. It's all a confusing pain in the ass, I get it. Until the pandemic wanes- which will happen when enough of the world is vaccinated- these contradictory and slightly illogical regulations will persist, I'm afraid.