I work for a pizza chain and we always just say cook.
Technically, the method of cooking is baking as that means "place in a hot chamber for a period of time". Other words, such as boil, roast, etc. have different meanings.
But people often use vocabulary appropriate to the type of food. They often talk about roasting meat because that is how meat used to be cooked, even if they are actually baking it.
Roast is related to rotate. Roasting meant turning something (usually meat) in front of, or over, the fire. Fat would drip off. This would be collected and poured back on. This is called basting. Potatoes etc. could be out in the tray that was collecting the fat, and thus get heated by the fire while being basted. Thus the term roast has been extended from "rotate" to "cook while pouring oil/fat over", or even "cook in a method that replaces roasting". Thus baked potatoes are baked without fat, but roast potatoes are baked with fat.
This is distinct from frying. In roasting the oil is there just for succulence, flavour, etc., but in frying the oil is heated and this is what cooks the food.
Bake is the traditional term for bread, cakes and biscuits, i.e. the things made by a baker. Pizza is a type of bread so bake is the correct term, however you look at it.
Some people may use terminology to reflect their particular process, e.g. fire to imply that their oven is particularly hot or contains flames.
It is also worth noting that a modern oven is heated while it is cooking, making it a much more flexible machine. Originally, you had to light a fire in the oven to heat it (so the oven was fired, not the food), then, when the fire was out, it was removed (so the food was not covered in too much smoke, ash and soot) and then the food was baked.