Long-time furry here. Let me try to break it down a bit.

"Furry" at its core is simply an interest in anthropomorphic animals. That covers a wide spectrum of people ranging from those that only have an interest in the art to therians who identify integrally as nonhuman. Whether you are a furry or not is just a matter of self-identification and whether you choose to engage with the fandom. I know incredible artists with whole-ass fursonas that don't call themselves furries and that's totally fine.

The puppygirl idea is something worth exploration though. Being into that doesn't necessarily mean you are a furry. There's an entire subculture of pup play that exists and while there is a fair bit of overlap between that and furry, it is a separate, distinct thing. Just food for thought!