So basically I should use "туловище"? Like for sentences such as 'that note is too high for him, you can see him contract his core very tightly' that kind of usage?
"Эта нота слишком высока для него, видно, как сильно он напрягает туловище (корпус)". To emphasize, one can say "напрягает все туловище" or "весь корпус".
Or "видно, как сильно он напрягает мыщцы туловища (корпуса)".
Just like in English, the Latin term sounds slightly more 'scientific'. But only if it really refers to the whole core, including abdomen, chest and back. Otherwise you can use more specific expressions, like "мыщцы живота" for abdominal muscles.
I feel the same way you do about loan words. I'd rather hear real Russian than loan words.
It's worse than loan word, it doesn't even exist in Russian. It's just that some people are too lazy to search for a word when they translate from foreign sources and then this non-word becomes sort of fashionable among aficionados