Here's the thing. You said a "pupper is a doggo."
Is it in the same family? Yes. No one's arguing that.
As someone who is a scientist who studies puppers, doggos, yappers, and even woofers, I am telling you, specifically, in doggology, no one calls puppers doggos. If you want to be "specific" like you said, then you shouldn't either. They're not the same thing.
If you're saying "doggo family" you're referring to the taxonomic grouping of Doggodaemous, which includes things from sub woofers to birdos to sharkos (the glub glub kind not the bork bork kind).
So your reasoning for calling a pupper a doggo is because random people "call the small yip yip ones doggos?" Let's get penguos and turkos in there, then, too.