I think the distinction you’re thinking of is when something is milled. Wheat is milled into flour. Rice is milled to remove the husks. Corn is milled into meal. At that point you’re not eating the plant, you’re eating a processed plant product. Of those three, corn is the only one that can really be eaten as-is, so perhaps the distinction of when it’s a grain or a vegetable is more about if it was dried and milled first.

But all of that seems unrelated to potatoes, which are roots. You can make bread out of potatoes, and I don’t think anyone would try to argue that potato bread somehow counts as a vegetable. But a potato on its own, minimally processed and eaten relatively whole, seems to fit the definition of a vegetable by most definitions, culinary or otherwise.