"First world" just meant the states that were America-aligned (so by definition, the U.S. goes here).

The "world" term is stupid and nobody should use it anymore (it implies that "third world" countries are lesser than America's allies)

Developing / developed economies is what more people use nowadays, and the U.S. would be classified as developed, though a binary is still not too helpful given there are lots of in-betweeens (the big elephant in the room, China, argues it should remain classified as a developing nation because, as it turns out, there is wealth inequality there too + billionaires taking too much of the pie, so a large chunk of people still live under conditions closer to developing countries, while others say that China should be a developed nation given they have the second strongest economy)

"First world" was a Cold War era term, where "first world" is America-aligned nations (so by definition, that would include the U.S.), "second world" was Soviet-sligned (no longer really in use since the Soviets no longer exist), and third world was everybody who didn't pick a side (so China after Sino-Soviet split, India, Brazil, etc.)

Most organisations use the more neutral terms "developed" and "developing", and the U.S. is definitely a developed nation (there are less binary terms that economists and such use to categorise even further that you could look into, just check the organisation who runs it since many can be biased towards certain things compared to others)

That's not the definition of first world... Switzerland is a third world country by definition