"Is Dima busy? Yes, he is doing homework."
Translation:Дима чем-то занят? Да, он делает уроки.
Indeed. If the creator of the sentence wanted чем-то to be part of the Russian translation, then perhaps "with something" should be part of the English sentence. Otherwise, the only way to know you're expected to translate "busy with something" is to translate just "busy" and get it wrong.
I find чем-то to be more fitting. reason being that you need to be busy on something that is not actually an abstract thing. he is working on something, hence why he is busy. no idea what he is working on, but we know it exists. As mentioned in the notes чем-нибудь is something hypothetical. that is more along the lines: this is unbearable, something needs to happen.