It can be exactly the same or if one is prone to overanalyze one could argue that: "We are eating" means "Vi er i færd med/gang med at spise" like in French "nous sommes en train de manger". "We eat" is a very general sentence, which can therefore take on a wider range of meanings. "Vi spiser", as in "We (actually) eat" as opposed to anything else. Or "We are eating" as in Why do you bother to ask. But to answer your question: You are right: one can use "spiser" in both cases.
Because they had to share it, they could only afford one, after all beef is expensive;) "a" is required because they are actually stating the amount "en" of steak, they are eating. One could write. "Vi spiser bøf/bøffer" i.e. with no article: "We are eating/having steak(s)" or "Vi spiser bøffen/bøfferne": "We are eating the steak/steaks"
Beef is not interchangeable with steak. Beef depends on the cut of the meat as to its name. A steak is from the most tender part of the animal therefore tastier. Roast tend to come from the rump in the arms or legs. Hamburger is ground meat. And that's just the start. Some people eat cow tongue which one cooked and peeled makes a nice sandwich. The French eat brains. Or at least you can order them in a French restaurant. There are a lot of parts to a cow or bull. Veal is a baby animal. A small calf. It's supposed to be more tender but on that regard I can't bring myself to eat a small animal I want it to grow up first. If you want a hamburger soup in Denmark order frigadella it's quite tasty
It's probably a case of outdated translation. There once was an expression called "beefsteak", which is highly misleading. It does not identify the part of the cut of the beef, which just means that we are talking about meat from grown cows/bulls. It is no longer appropriate to use the word "bøf" in Danish as an identifier of anything in particular. It at least needs a qualifier as e.g. "Engelsk Bøf." Nowadays everything is sold with a bit more precision, but not entirely. For instance "En mørbradbøf" (mørbrad is the tenderloin, which is the most tender piece of the animal) can be pork as well as beef. Therefore there is a generally widespread practice (like in the rest of the world) of using specifics as in "svinemørbrad, oksemørbrad or lammemørbrad" i.e. "pork tenderloin, beef tenderloin or lamb tenderloin."