I don't know why they wouldn't call a 'фітнес-зал' а 'fitness-gym/club'.
It's more specific, as there are different types of gyms.
You are right. In Ukrainian,
gym -> спортзал, спортивна зала
fitness center -> фітнес-центр
I don't know why they are used interchangeably here...
I don't get what зал means
It means "hall", so fitness hall.
Think of the German word Saal, which also means hall, considering you are learn German.
Do you really use фітнес - зал expression in Ukraine? It sounds strange to me as Belarusian
I think people tend to say "спортзал".
Why gym and not gymnasium?
Good point. Added.
His would be "that gym"?