No, cake doughnuts are also fried. “Cake” doughnut simply means they’re made without yeast but with a cake batter using baking powder or baking soda as leavening, so they have a cakier texture than a yeast-raised doughnut.
Wikipedia:
There are two types of ring doughnuts, those made from a yeast-based dough for raised doughnuts, or those made from a special type of cake batter. Yeast-raised doughnuts contain about 25% oil by weight, whereas cake doughnuts’ oil content is around 20%, but they have extra fat included in the batter before frying. Cake doughnuts are fried for about 90 seconds at approximately 190 to 198 °C (374 to 388 °F), turning once. Yeast-raised doughnuts absorb more oil because they take longer to fry, about 150 seconds, at 182 to 190 °C (360 to 374 °F).
There is such a thing as doughnuts that are baked in the oven instead of being fried. Those are called “Baked Doughnuts”. They’re usually cake doughnut batter, baked in a doughnut pan. Baked yeast-raised doughnuts seem to be a little less common, at least for home cooks, as they’re more challenging to make.