That IS confusing - I hadn't ever read it.
Yep, lifting weights = weight training = weight lifting = strength training = toning ... they're all the same thing.
If you're training for strength, you do higher weights and lower reps. A rep range of 8 - 12 is good for gaining strength (you want to pick a weight so you're pretty close to failure before you reach 12). If you're training for endurance, you do high reps with lower weights. I'd call anything in the 15+ rep range endurance training.
I like to mix up both endurance and strength training in the same workout. Let's say I'm doing BB squats ... I'd start out with the 45# bar to warm up, go to 15/95, 12/115, 8/135, 6/155, and finish off with 25/95. The first two sets are warm-up, the next three are for strength, and the last set is for endurance. Or you can do strength and endurance workouts on different days, like the article is talking about. There isn't any 'right' and 'wrong' way to do it.
I hope that makes it a little clearer and not more confusing!