This is probably not the correct definition, but when I think of campy as it related to Star Trek I think about some of the acting in certain moments (Kirk pushing out his hands as he switches places with Lester, screaming "I'm Captain Kirk!!" in 1.5), the lighting on the female guest stars and the music that plays when it shows them, some of the fight scenes). I don't consider TOS a campy show - it is a scifi/drama show and holds up that way, but it does have some campy moments to me..but that's because I'm watching it in 2022 as opposed to something that was intended to be that way.
Yes, there's a distinction between intentional camp (Adam West's style on Batman and Family Guy is the supreme example) versus serious, stylish work that was done in all sincerity, but can be laughed at decades later because of changes to filmmaking that happened in the meantime.
I always took the "beautiful woman with soft lighting and a love theme" seriously on Star Trek, I ate it up with a spoon, and I was eager for more when Moonlighting gave Cybill Shepherd the same treatment in 1985. It wasn't a joke, it was a re-imagining of life as it ought to be, rather than what it is usually is. (Moonlighting would test the fourth wall and make fun of itself once in a while, especially late in the series when the show was falling apart and they were going for broke, but Cybill's beauty shots were rightly sincere.)