self-made vs. home-made

The English expression self-made is quite often incorrectly used as a translation of the German word selbstgemacht. A classic example is

I brought self-made cake to work today. (Wrong)

Interestingly, the situation is more complicated than I expected. In particular, whether or not self-made is a false friend (and should be replaced with home-made) depends on the context. According to the online version of the OLD, we have the following meanings:

self-made: having become rich and successful through your own hard work rather than having had money given to you

home-made: made at home, rather than produced in a factory and bought in a shop/store

However, according to several online discussions I found, there are exceptions. Moreover, self-made is not included in any of the lists of German-English false friends I consulted. Turning to my trusty old Collins German-English Dictionary, I found the following explanation: the German word selbstgemacht translates to self-made when talking about furniture or other equipment, and to home-made when talking about food, drinks, etc. Therefore, a cake is never self-made, but it might be home-made. A good way out is to refer to whatever you made by saying

I made it myself.