What is this part of the cake called?

Somehow, this all recalls the Seinfeld episode where Elaine went into the muffin top business- the difficulty being what to do with the "stumps".

I like that part too, and also cake trimmings for the same reason - they're sort of a baker's treat! Not sure if you can make something that is just like that left behind cake, but a crepe cake might come close to a the moist top part of cakes that are usually trimmed off. It's not what you're going for but hear me out, I bet you'd enjoy a good sticky toffee pudding (or similar English style pudding/cake) - I made a recipe from David Lebovitz, and smothering a mini cake in sticky toffy basically increased the area of that super moist layer on the outside of cakes plus the inside was wonderfully tender.

Are you talking about the filling? For some cakes, a different confection is made to spread between cake layers than the frosting that is spread or poured on the top and sides. This can be a custard like crème patissière, a mousse, stabilized whipped cream or a citrus or other curd. It affords a textural contrast to the other components and enhances or highlights the flavors. Sometimes a cake layer is sprinkled with liqueur to add flavor and moisture before the filling is spread on. The bottom layer will absorb the moisture from these treatments and perhaps become a shade darker. I don't know that there is a specific term for the bottom layer, though, other than base or bottom layer. You can make a simple, elegant one layer cake, such as Maida Heatter's lovely "Queen Mother's Cake" (from "Maida Heatter's Book of Great Desserts", just google Queen Mother's Cake for the recipe), with a poured ganache topping--no layers, no filling), perhaps moistening first with raspberry or other liqueur if you wish.

It sounds like you're describing the portion of cake that is left behind in the cake pan after baking, or a cake plate. This is just the "skin" or outside of the cake that has been browned by the oven. Some bakers cut it off before icing to have more uniform layers. You should try baking a cake to see for yourself as an experiment.