'Historically' not owning a TV was really just saying that you don't watch broadcast television and, by extension, that you don't really watch much TV (like The Onion's Area Man [1]).
I agree that this interpretation doesn't make sense anymore. What fascinates me is that many people will still use "I don't own a TV" to mean the same thing. Even more interesting is that more than a few people I know, myself included, will even believe what that statement implies despite it being completely inaccurate.
It wasn't until recently that I realized that not having a TV had absolutely no impact on my consumption of entertainment, and yet somehow it still felt like I was being 'more responsible' by not having one. Nevermind the fact that I'll happily binge on Netflix or watch hours of YouTube, and not just in my living room at broadcast times, but anywhere and at any time where I have 'nothing to do'.
Maybe one day people will brag about having only one dedicated media consumption device at a fixed location in their house!
EDIT: I just realized that another common implication of "I don't own a TV" is that the person is more discerning in their consumption and only watch 'good stuff' (HBO, feature films, etc.). Having spent loads of time watching only the highest quality shows (of which there many nowadays), I can't help but feel that this is also very often just another justification to consume relatively pointless media that just happens to be well-written and expensive. Not that there's anything wrong with that.
[1]: http://www.theonion.com/article/area-man-constantly-mentioni...