Whenever it comes down to definitions I like to go to expert definitions rather than common language. For food (are tomatoes a fruit?) I use FDA definitions, for which the definition of bread excludes what you'd mean by "cake".
- Layman readable synopsis (bakerpedia)
- Sec. 136.110 Bread, rolls, and buns. (fda)
- Sec. 136.115 Enriched bread, rolls, and buns. (fda)
I don't think the FDA defines cake, but it does specify how different types of cakes, brownies and such should be labeled (search for "cake" here).