Paprika is a type of chile pepper. It’s a type of Capsicum annum. I have seen them grown in Szeged and Kalocsa. You see wreaths of them around the houses of the villages in those areas during early fall. The sweet kind is made by removing the seeds. The hot kind is made with the seeds put back in. And there are varieties in the middle (bitter-sharp), and a whole mess of other kinds. There’s at least a half dozen different designations of paprika at the local markets. I can think of seven: special, mild, delicatess, sweet, semisweet, rose, and hot, plus you have some smoked varieties as well. The hot ones (depending on brand) can get to around chile arbol levels of heat, somewhere approaching medium-hot. But the pepper itself is absolutely a chile pepper.
At any rate, many other chiles are grown in Hungary. There is one that goes by the name magyar erõs paprika (“Hungarian strong”), which is similar to perhaps an anaheim or New Mexico chile. There is bogyiszloi, which is spicier, and looks a bit like a bigger version of a fresno pepper with pale yellow-green skin. There is also the cherry pepper, which is often dried and used at the table to spice up food. It can also be found pickled. There is almapaprika, which is a mildly spicy pepper that is usually served pickled (and my favorite accompaniment to a Hungarian stew.)
Yes, most Hungarians do not cook their dishes to hot levels. But you can certainly find stuff I would consider spicy. Fish soup, for example, tends to be served on the spicy side. And with any of the typical dishes, you will usually find a jar of Erós Pista (“Stong Steve” – a hot pepper paste similar to sambal oelek) or dried hot chili peppers for you to crumble over your dish as you see fit. But go to the countryside, and the food does start trending spicy.