Here's the thing. You said a "dolphin is a whale."
Is it in the same family? Yes. No one's arguing that.
As someone who is a scientist who studies whales, I am telling you, specifically, in science, no one calls dolphins whales. If you want to be "specific" like you said, then you shouldn't either. They're not the same thing.
If you're saying "whale family" you're referring to the taxonomic grouping of Cetacea, which includes things from porpoises to belugas to orcas.
So your reasoning for calling a dolphin a whale is because random people "call the big ones whales?" Let's get whale sharks and great whites in there, then, too.
Also, calling someone a human or an ape? It's not one or the other, that's not how taxonomy works. They're both. A dolphin is a dolphin and a member of the whale family. But that's not what you said. You said a dolphin is a whale, which is not true unless you're okay with calling all members of the whale family whales, which means you'd call vaquitas, bottlenose, and other marine mammals whales, too. Which you said you don't.
It's okay to just admit you're wrong, you know?