Someone else used the term "free climbing", but you're right, I've too have always understood it as climbing without rope. What I'm talking about is light climbing, where people leave the overnight gear behind for a summit, preferring to keep to a small pack for speed and hope to make it up and down without some calamity requiring an overnight stay. Summitting high peaks without oxygen is another parallel with stupid diving as it's more dangerous to forego the use of oxygen just as it's more dangerous to forego the use of helium on deep dives.
Free climbing is when you climb without pulling on or using anything other than your body to get up the rock. This is contrary to aid climbing which is when a climber puts gear in the rock and literally pulls up on it to accend. Free soloing is with no aid and no rope. If you want a climbing analogy, then it needs to be this:
A sport climber (a free climber who only fixes the rope to pre-drilled protection points) decides to embark on a long multi-pitch route that requires he/ she place all of their own protection/ set up all anchors/ and do a route that requires route finding. Additionally, this sport climber does this style of climbing not well within their own established limits in sport climbing, but does a route at or near their sport climbing limit. The result is an ill prepared climber who will not be seen as pushing the limits of their sport by their peers, but someone who ignored the reality that their experience in one aspect of the sport does in no way prepare them for any other aspect of the sport.
Additionally, climbers who climb at extreme altitude with no oxygen, are able to accomplish this with dedicated training. Again, speed in the mountains adds an element of safety. Climbers who go fast and light, still use safety gear, they just use less of it, and they are well trained to do so.
I am not trying to turn this into a climbing thread. Nor am I trying to nit pick about climbing definitions. I see parallels between climbing and diving, and I'd like the parallels to be accurate.