Baking bread is more correct, but in my experience "to cook" is generic enough to include baking. If someone has something in the oven and I ask "what are you cooking?", it's not weird.
On the other hand, "cooking bread" is like 2/10 weird.
Even if you assume literal translations capture 100% of the details, "kochen" has multiple meanings. One of them is "boiling", another is "cooking" in the general sense of "preparing meals".
Well Wikipedia says there is a narrow definition, that necessarily involves boiling of a liquid. This is the one I always used.
But in the broader sense, it seems to also mean the preparing of meals, but I never encountered it like this.
"Kochen (von lateinisch coquere, „kochen, sieden, reifen“ entlehnt) ist im engeren Sinne das Erhitzen einer Flüssigkeit bis zum und am Siedepunkt, im Weiteren das Garen oder Zubereiten von Lebensmitteln allgemein"
What's the difference between baking and cooking in your understanding? You can make bread in a rice cooker, which nobody has ever called a "rice baker".
edit: and a rice cooker is definitely not a small oven.
And the german definition of cooking in the narrow sense is defined like this, but in a broader sense apparently usable for everything with preparing meals.
And I never used a rice cooker, so no idea how to classify that ..
If cooking involves boiling, what are you doing when you put a steak in a hot cast-iron pan?
To the rice cooker point, I'd argue that an oven uses a heating element of some form (electric coils, gas flame, wood fire) to heat the air in a closed environment, and the air transfers heat into an item. In contrast, a rice cooker uses a heating element to directly heat a metal pot, and the metal pot transfers heat into an item. Usually that's going to be a combination of rice and water, but you can e.g. pour pancake batter into the pot and get a large souffle pancake, or put bread dough into the pot and get a loaf of bread. The trick is that the metal pot is much more efficient at transferring heat than the air is, so the rice cooker doesn't need to be at the same temperature as an oven to get the same amount of heat into whatever you're cooking.
Well, technically there is usually bubbling going on, when making a steak, but would you "cook a steak" in english?
In german you would not, one would roast it. (but we have 2 words, roasting "rösten" on the bbq and "braten" would be in a pan. But a "Braten" would be in an oven.)
Kind of not that consistent (like it usually is with natural language).
In general I think those terms were invented, before there were things as a rice cooker.
On the other hand, "cooking bread" is like 2/10 weird.