Oh, well thanks for being open minded. There is definitely a perception with a lot of people that furry is just sexual. And like with anything made by humans there is a sexual component to the fandom. But that is more because it is a very open and accepting group of people. So for the most part sex is seen as something normal that happens rather than some shameful bad thing. So there are about an equal number of furries who appreciate NSFW art and just as many who aren't interested in it.
I did find it really interesting that many people in this thread mentioned furry as an art style. There is definitely no 1 furry art style. You can draw anthro characters in any style you please and there are hundreds of thousands of furry artists who all have their own unique style they bring to what they create. Like with anything there may be trends that come and go in what kinds of art styles are popular, but this fandom has been around since the 70's and the favored art styles have shifted so much over the course of 50 or so years.
But to your question of
Non-anthropomorphic animals that still have humanized facial expressions and minds can still fall under the furry umbrella. Think like the lion king. Typically these characters are called "Feral" in the fandom, which just means "physically non-anthropomoric" aside from facial expressions. But like I said in my previous post the creator may not label themselves or their work as furry at all. Even if it is enjoyed by lots and lots of furries.
For example, I would classify this comic as under that Feral furry umbrella. Just as a reader. I have no idea if the author would say the same.
Or like this novel by @Phantomlight
All of the wolves in their story have essentially human minds and emotions with a bit of animal body language added in.
