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.
So reading up on the evolution of whales for arguments sake has me realising all dolphins and whales are (as mentioned) from the same family.
Your traditional whale fits into "Baleen Whales (Mysticeti)" which have "soft, hair like structures on the upper mouth" and there are 16 species and 3 families.
Meanwhile there are also "Toothed Whales (Odontceti)" with 76 species and 10 families. They are smaller, actively hunt and almost always live in pods.
The most surprising thing I've learned is that the Baleen Whales typically have two blow holes...??? Also they do not echolocate but they do sing/chat.
So almost all your traditional large whales fit into the Baleen category and the traditional dolphin fits into the Toothed category. So there are key differences between them, but the overall family is whale.
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 you said you don't.
It's okay to just admit you're wrong, you know?
You lost me here. OP never said dolphins aren't dolphins. They also never said other non-dolphin members of the whale family wouldn't be called whales. Where are you getting that from?