Frankly, most personal trainers here in the US are complete frauds. They pay $300, study a few hours for a test, and hang out their shingle. They read up on the latest fads in exercising, and advise their clients accordingly.
The bunny dumbbell stuff while doing cardio started decades ago. A lot of hucksters promoted it as a "magic" way to lose more weight, supposedly because you increased the resistance. In reality, what you're usually doing is putting stress on bones and joints that's completely unnatural -- and that's what keeps the orthopedic surgeons in business.
Aerobic = oxygen. In other words, your muscles are getting enough oxygen to function without going into their glycogen stores. Holding a couple of 5 lb. weights won't change that one iota, but it CAN put a lot of repetitive stress on your wrists and your elbow.
Perhaps the personal trainer believed that when you held these little weights, somehow that would strengthen your quads and your knees. I guarantee you -- if you want to do that, squats will accomplish that much more effectively and safely.
One of the biggest problems with mixing cardio and weights is form. The weights you're carrying tend to ruin your cardio form, and the cardio tends to ruin your weight lifting form.
If you don't believe how difficult it is to have proper weight lifting form, sit on a kitchen chair and try to do a proper military press, with perfect control, even using the bunny weights you already have. You'll probably find your arms bowing out on the way up, the weight shaking back and forth, too rapid a descent, incorrect breathing, etc. etc.
Form is absolutely critical with weight training. It takes concentration. Anyone can do it, though. It just requires focus.
So with a treadmill at home, a bench, and a couple of 15, 20, or 25 lb. dumbbells (you'll have to figure out the right ones to start with), you can design a perfectly good cardio / weight training routine that is both safe and effective.
By the way, here's a nice little video on the difference between aerobic and anaerobic exercise: