If you're goal is fat loss, not cardiovascular health, then lifting weights works just as well as running or using the elliptical, and it is easier on your body if you learn the correct form.
Of course you said fat loss, and perhaps you'll quibble about how much of the calories burned comes from fat vs. muscle, but I think more people are trying to get to a healthy weight rather than achieve a single digit body fat percentage.
When people say healthy weight surely they always mean healthy body fat percentage?
The point is that anaerobic exercise builds muscle, which boosts your basal metabolic rate. Unless you're doing cardio several hours per day, lifting will be much more efficient.
edit: minor word change for clarity.
With resistance training you are not trying to burn calories, you are building muscle. This new muscle will need calories to support itself and will make it easier to lose fat in the future because your body will need more calories to function.
I've done that a few times. Consistently 140-160 BPM, peaking higher, sustained over 40-80 minutes.
My peak during Tabata sets is 180-190 BPM, so lifting isn't quite at that rate, but it's decidedly aerobic, especially for mid-high range sets (5-20 reps).
That's a big if. I don't trust myself to do it properly by myself based off a diagram from the internet. What's my alternative? The evil big box gym and overpriced personal trainer?
Or get Starting Strength, by Mark Rippetoe, which has about 70 pages of instructions and diagrams on the correct form of each lift.
Or find a gym that has a personal trainer willing to teach you the proper form.
Or do it, record yourself with a camera of some sort, and find a community of weightlifting people (/r/fitness welcomes this) to correct your form.