Broadly speaking, there are 2 types of COVID tests available :
Those carried out at a testing provider, where the test itself is either done by the provider, or by the patent under the instruction of the provider. In such situations the provider will have checked the patents identification (and potentially insurance details, etc) before taking the test, and that name associated with the patent will be included in the test results. This includes the 'drive-thru' tests you mentioned.
"Home" tests, where the test is completed entirely by the patent at any location.
The requirements for testing before travel to the US are relatively low (compared to many other countries), and many forms of "home" are accepted by the US - specifically "antigen" tests (sometimes also referred to as "Rapid" tests).
However the problem with home testing is that there is generally no way to show verified results to the airline or the US immigration staff to prove that the test was negative, or even who the test was carried out on.
In order to overcome this, several companies offer "proctored" testing services, where you connect to a representative of the testing company via a video call, show them your identification, and then carry out the test (and wait for the results) with them watching via the call. Once the test results are available, they will provide certification of your test results, including details of who the test was taken by and the time it was taken - which are the details that the airline requires.
For travel purposes, EITHER of these options is suitable. ie, testing carried out at a testing center (regardless of who physically administers the test), OR testing at another location with the test visually proctored by a remote agent. In either case, what matters is that you will require documented results showing the type of testing performed, the time of the test, and the testing subjects name. Some testing locations will NOT provide this level of detail for standard testing, and may require payment of a fee to do so - if they offer it at all. For example, in some areas sending negative results via a SMS message is common, and is generally NOT acceptable for travel purposes.