Is it though? Because the orgins of the word refer to the inner side of a record that's less likely to get played. And if you own a record, are you not a fan?
Sometimes people don't own a record, maybe they only hear singles, only bought the single/7", only stream a playlist or don't know of non-single tracks. I think depending on the body of work, a song that was a single can be a deep cut. It could have been minimally popular and then fallen off, so a single or something that was once on the radio can be a deep cut depending on the context.
I think it also has to be an album track. A single released only as a b-side, for example, is just that. A b-side/rarity. A deep cut is an underappreciated album track.
It’s neither but it can also be both. It’s by very definition a song that you have to scratch below the surface to discover. What that is can vary massively depending on the act and the song.