...otherwise, we'd call a 1954 Beetle a coupe, no?
The whole concept is muddied and was never particularly clear in the first instance.
For a long time, it has just been used to describe 2-door with ‘sporty’ lines. Which could be interpreted as just as much a ‘marketing’ use as applying it to a 4-door.
Is an E21 BMW a coupe? Yes, it is a 2-door, but what was 'cut' to make it a coupe? Nothing. Hence, it should be a 2-door sedan. But that sounds unsporty and boring.
Back when you had a big, upright 4 door sedan like an S1 Bentley, Mulliner would build a whole new body for you, removing the rear doors, lengthening the fronts, making the back seat less accommodating via a swoopy roofline....and call it a Continental. THAT was a Coupe. A 2-door, "cut" version of the 4-door sedan. Simple.
Even the Rover P5 "coupe" has a better claim (in some ways) than ‘just’ a 2 door sedan. Rover took a the P5 4 door sedan and hacked down the roof-line for a swoopy c-pillar, removing a lot of headroom from the rear. Cut, indeed.
And a better claim than BMW 4-series Gran Coupe, which doesn’t have much of a different silhouette to the 2-door 4 Series. I’d actually designate the Gran Coupe as a 4 door hatchback. Unsporty and boring, again, not to mention plebian...not good for the marketing department.
Next up: Definitions of Drop Head Coupes versus Coupe Convertibles