What activity am I really doing?

So my husband and I have been working with a friend at the gym who offered to train us. We both need to drop some serious weight and we're totally thankful for our friend who's coaching us along and helping us out. My problem is knowing what I'm burning each time. Right now our workouts consist of 20-25 minutes of cardio (bike or elliptical) and then 45-60 minutes of weights. Our weight sessions consist of 4-5 exercises or machines, in a circuit. 5 sets, 20 reps, usually starting out with a warmup weight and increasing if needed on sets 2-5. On a good day when we don't have to wait and our friend is switching the weights for us, I get moving with minimal stops, definitely have to catch my breath, and feel my heart rate increasing, but not as much as I would assume with an actual cardio workout.

Initially, I was logging this as strength training, which has a pretty low calorie burn. My trainer looked at our diaries and told us to log this as circuit training, as we were working a circuit. When I logged that for the first time, it was over triple what strength training is and I netted negative calories for the day. While that was cool to see, I didn't feel like I burned that much, even though I am at a weight level of morbid obesity. Right now I'm logging 25% time Circuit and 75% strength.

I thought the obvious answer was a HRM, but the more I read, the more I see that HRMs aren't going to do me any good for weight training, only cardio. Any thoughts on what I should log this as? Would a HRM do me any good? This might be an impossible question, but I was just hoping for any feedback I could get.