I’d say cakes are all bread, but not all bread is cake. Likewise, I’d say omelettes are a type of egg dish, as are plain scrambled eggs, but not all egg dishes are one of those.
If you kept to Western cuisine you could argue bread as a distinct category both within “homogeneous baked goods” or something, but then ingera (for example) would probably end up being a cake, and that’s not quite right. It’s more important that bread include all solid grain-based staples the world over than that it exclude Western cake, IMO.