"Skipping rope" vs. "jump rope"
Here in America, jump rope is the only noun we use. At my gym we warm up by jumping rope so I own my own jump rope of just the right length and thickness. As an adult wanting to exercise, I jump rope (doing single unders and double unders) at my gym.
Young girls at play might jump rope or skip rope or even play jump rope. Skip rope implies (at least to me) a lazy, fun activity one can do while daydreaming, while even young girls might jump rope with a sense of competition. But even girls who skip rope do so with a jump rope.
Before I read this question I don't think I'd ever heard the term skipping rope used as a noun to mean the rope.
This is an article on using a jump rope as part of an athletic training regimen that is written in American English with an attempt to include British English terms as alternates. I can't speak to the British, but the American seems natural and correct to me.
I didn't know Americans called a skipping rope something different. I'd be very surprised if it's ever been called anything else in British English (I don't have access to the OED at the moment).
One meaning of skip is "jump". CDs can skip momentarily, meaning that the read head has jumped from where it should be.
However, skipping is also that sort of running, jumping, playful walk that children like to do. And it's possible to do that sort of skipping on-the-spot with a skipping rope — you don't simply have to jump up and down as you use it.