I was wondering if the part of the network with frozen weights has a specific name. Similarly, I'd want to know whether the custom-made part also has a specific name.
I don't think there is any scientific terminology for the part of the model, however, researchers do use the terminology for the tasks and domains for the pre-trained and custom models being used for transfer learning - Source and Target tasks / domains.
More information to this definition can be found in the IEEE paper A Survey on Transfer Learning (Sinno Jialin Pan and Qiang Yang)
Side Note: In my personal experience though, in industry, we usually call the Source task's model as the Source model / pre-trained model, and the Target task's model as the downstream estimator / downstream model.