So I made a contract with a gym to lose fat and become fit. My goal is something like this 
Now I'm over weight so I need to lose fat (I think something like 14KG). I told the trainer in the gym about my goals and that I don't want big muscles or so. So the program he gave to me is to do exercises with some weight machines in the gym:
1- a machine that trains the legs muscles.
2- another the trains the back, and chest.
3- then an exercise with TRX for my stomach.
& that was it. Then I was so confused that he didn't assign me any cardio exercises like running machine or cycling! His argument was that when I train my muscles, then the body needs to burn fat and that's how the fat will be burned. But then I insisted to go on the running machine and then he asked me to run one minute fast and then one minute slow. I found that I had a lot of sweat with the running machine than with the weight machines!
The trainer said that I should do many repetitions with low weights on the machines and that's how I achieve my goal.
My question is:
1- Is the trainer correct? Is it better to use the weight machines for fat burning than with cardio machines?
I'm worried because I don't want big muscles! I want to be fit and lose fat quickly! I don't want to work hard and then find out that it's not the bodyshape that I want.