For TV you count the height, as that's the only thing that matters — you can always add more high-frequency signal to increase the width after all. That's why TV signals for e.g. DVD also use a 480p signal for 4:3 and 16:9, you just stretch the content, no problem on analog TV. (In fact, the "p" doesn't mean pixels, it means the content isn't interlaced (progressive) while 1080i would be interlaced content, with both 540p and 1080i having the same amount of lines per frame)
While for cinema you count the width, e.g. what's more commonly known as 1080p is just 2K in cinema terms, as it's 1920 (or roughly 2048) wide. And that's why 4K is 4K, as cinema content is actually rendered in 4096 pixels width. In cinemas you can always add or remove height (just don't project onto that part of the screen, some cinemas even have screens able to change their height), while the width is fixed (you usually can't make the screen wider). Historically this also held true with film: except for IMAX (70mm portrait), you measure the width of the film e.g. 8mm, 16mm, 35mm, 70mm, as you can always just make the film longer to change the height.
It's true that manufacturers quickly started calling their UHD content 4K for marketing reasons, but 4K doesn't have anything to do with your hypothesis.
Now why was it used as marketing term? When HD and Full HD rolled out for consumers, cinemas started upgrading their projectors as well, offering content in 4K, and advertising it as such. Consumers got used to 4K meaning high resolution high quality. And when consumer devices started reaching a similar quality, obviously the manufacturers used the already established term, instead of the official "UHD" branding.
720P was branded HD, 4K was branded UHD, but 1080P never got a designation like that. This put Sky in an interesting position, as they only recently added 1080P support to their NowTV streaming subscription service, as an optional upgrade to their usual 720P. Having already long promoted their service as delivering HD, [0] they were clearly at a bit of a loss as to how to describe 1080P. They ended up calling it full HD. [1]
[0] https://community.nowtv.com/t5/Sports/HD-or-non-HD/m-p/40699...
So it's not like sky invented that term.
Of course I still don't forgive them for decimal disk capacities (in my book 1 meg is still 1048576 not 1000000)
If this is HD:
┏━━━━┓
┃ ┃
┗━━━━┛
┏━━━━┳━━━━┓
┃ ┃ ┃
┣━━━━╋━━━━┫
┃ ┃ ┃
┗━━━━┻━━━━┛
640x480 ("480p" <-- vertical)
1280x720 ("720p" <-- vertical)
1920x1080 ("1080p" <-- vertical)
3840x2160 ("4K" <-- they marketing departments suddenly started measuring horizontal)
720p, 1080p came from the TV industry since they always counted video in "lines" (NTSC is 525 lines)
4K came from the movie industry (the 2005 Digital Cinema specification), since movies have wildly different aspect ratios so counting the vertical doesn't make sense (specifically, it's very rare for movies to be 16:9!)
4K > Full HD > HD > SD (marketing terms)
2160p/i > 1080p/i > 720p/ > 480p/i / 576p/i (resolutions)
Displays used to be 480p, 720p, 1080p, and then ... 4K?
The 4K was clearly intended to fool people into thinking that it was 4Kp or 4000p.
But get a load of this... 4K is 2160p. It should have been called 2Kp to be a fair comparison with the then-state-of-the-art 1080p.
The marketing departments quietly went from measuring the vertical to measuring the horizontal to fool a bunch of unsuspecting consumers.