Broadly speaking, the economy “main” cabin of typical domestic airliners has a few rows at the front that have the same 6-abreast seats, but a couple inches more legroom. Works well for taller, but not wider, people. That’s “premium economy”, “main cabin extra”, etc.
Airlines like Frontier and the late unlamented Spirit invented ultra cheap fares but where everything, including carryon luggage and picking your own seat costs extra. Gives a really great headline price but once you add “fries with that” it adds up to almost the same price as The Other Guys.
In response to Frontier, Spirit, et al, the majors began offering “basic economy” or similar names. Which is the ordinary coach seats, so much nicer seats with more legroom than on Frontier, but the same no-frills, everything costs extra al la carte approach to pricing. That way they appear competitively priced w Frontier when folks are searching flights on Expedia, TripAdvisor, Kayak, etc. The majors would love for that category of fares to go away. The recent death of Spirit is a promising omen in that direction.
Which leaves “classic” or similar names for the ordinary middle-of-the-road coach seating & included amenities.
As to “first class” and especially US domestic “first class” …
At least domestically, the front few rows with 4-abreast on narrowbodies and lots of legroom have not been called “First class” for about 30 years now. That’s “business class”. Why? Because in the 1990s there was a corporate fashion to not allow their employees to book “first class”. Sounded like wasting valuable shareholder profits on luxo perks for the execs and middle managers. No more “first class” for them.
So all the airlines almost instantly rebranded the same product either “business class” or “premium class”. Problem solved. The execs and frequent fliers and middle managers on expense accounts still sit up front, the shareholders and Wall Street analysts went away happy that they’d wrung more profit out of the company’s employees, and everybody won. Or thought they did.
Long haul international flights on US carriers and both short- and long-haul on non-US carriers’ practices are different yet again and more complex.
For example, last week I rode transatlantic on an Air France 777. There are 4 distinct levels of physical seating arrangement and doubtless about 2 dozen different fare categories for those seats: Seat map of Air France Boeing 777-300ER aircraft.
For anyone buying any air travel on any carrier, this website is a very valuable resources to separate the fare mumbo jumbo from the actual physical seating arrangements.
I’d rather plan vacations, oh well.