Weight lifting is rarely a sustained form of heightened heart activity in the way the other examples are. Yes it works your heart, but there are usually many relatively motionless pauses between short sets, as opposed to the constant push on your heart that accompanies more traditional forms of cardio work. Not a knock against weight lifting, just emphasizing a different type of heart activity.
Aren't those pauses usually people standing around trying to catch their breath, with their pulses coming down from the 180 they hit while doing a set?
Isn't weight lifting more like doing a 40m sprint, then slowly walking back to do it again?
My recommendation to the OP, since it was asked, is to engage in sustained cardiovascular activity, where the heart rate is elevated at a consistent rate. I'm not saying weight lifting doesn't have its strengths (no pun intended!) but it's not the type of cardio activity that I'm suggesting, for the same reason as what you've just mentioned: the idea is not to push your heart to an extreme and let the pulse "come down"... it's to get your heart to an elevated rate and keep it there consistently for a long stretch. For that type of activity, weight lifting is not the answer. For other types of goals, it very well may be.
Different types of cardio workouts definitely exist, and most trainers will tell you that a well-rounded health regimen requires a mix of many different types, including weight lifting juxtaposed with more cardio-centric exercises. But this thread is not about how to give your body the best workout.