Here is a list of the national affiliates of the International Jump Rope Union: https://ijru.sport/members
Their names presumably give some indication of how the activity is generally referred to in various places, at least in official contexts. The only one that includes the word "Skipping" without mentioning "Rope" is Australia's (although there is no UK affiliate listed).
I don't know what you find misleading or tautological about the term. If you only used "skipping" to refer to an activity with a rope, then I suppose "skipping rope" would be tautological, but you previously mentioned that "skipping can mean using a rope or propelling oneself happily in any direction." Other speakers distinguish these activities lexically. For them referring to the first simply as "skipping" would likely lead to misunderstanding.