How do I log this?

Okay -- I need some help you guys. I've recently switched up my workout routine. I'll try and make it as simple as possible when I explain it:

Before I go to the gym, I look through my exercise diary to see what set of machines I should be doing that day (due to a shoulder issue, I have to use machines vs the weights). For example, I might use the arm curl machine, the tricep one and the chest press. I track the weight I use and usually change my weight once a month or so. My goal is 10-15 reps on each machine.

Once I get to the gym, I warm up for 10 mins on the treadmill - 5 walking and then 5 running at a medium speed (like 4.6). Then I go do each of my 3 machines. Then back to the treadmill for 10 mins of running at 5.0 or so. Then back to the 3 machines again. Back to the treadmill for 10 mins of walking (3.8 or 3.9) with the incline set at 5 or 6. Then back to the machines for the final time. I then do each of the hip machines for 10 reps. After that, I head back to the treadmill for lower speed running (4.3) for 5 mins and then I cool down.

Should I be logging this as circuit training? I know I keep my heart rate up the entire time because I check it while on the treadmill each time I get off the treadmill, and then when I come back to the treadmill after doing the machines.

Unfortunately, right now a HRM isn't in the budget, so I'm doing the best I can with what I have to work with. I've been logging the number of calories the treadmill says I burn and then logging the machine time as "strength training". Someone suggested logging it as circuit training since my heart rate is up the entire time.