A quick Google search suggests to me that not everyone agrees on what is meant by alpine climbing. I think of it as climbing high in the mountains, usually in a somewhat remote area, and at least a little bit technical. Personally, I would not consider a class 1 or 2 hike up a fourteener in Colorado in the summer as alpine climbing. The same thing only class 3, maybe; class 4, probably; class 5, definitely; and Mount Rainier, definitely. Rock climbing at your local crag or ice climbing at the Ouray Ice Park does qualify. Your definition might be different.
My experience with alpine climbing is very limited. I would like to learn and do more.
I climbed Vestal Peak in Colorado via the Wham Ridge with my son leading. A brief report of that trip can be seen here: Trip Report: Vestal Peak Climb, Wham Ridge Route, 2006. The route description can be seen here: Vestal Peak, Wham Ridge Route on Mountain Project.
I attempted Mount Rainier via the Kautz Route guided by Rainier Mountaineering Inc. Because of weather, no one in the group of 8 clients and 4 guides summited. At that time, even in good weather, I did not have the physical ability to complete the climb.
Sometime in the mid-late 1960's, I was hiking with my father near Silverton, CO. We were climbing out of Lower Ice Lake Basin to Upper Ice Lake Basin. The steep ramp was still completely covered in snow. I do not remember what month we were there but probably June or July.
I climbed up the ramp and belayed my father with a boot/axe belay. He fell, penduluming off the side of the ramp. Amazingly, the belay held fast. We had the rope tied around our waists, no harnesses, no crampons. We had no idea what we were doing.
He easily scrambled to his feet and back up onto the ramp. We immediately aborted our plans to ascend into Upper Ice Lake Basin and hiked down to the South Mineral Campground Trailhead.I used a wooden-handled Chouinard ice axe to make that catch. I do not have it anymore. It was stolen from our condo at Silverpick near Purgatory Ski Area when our condo rentals were managed by Chris Wing. It was on display above the fireplace mantle along with the Goldline rope and Grivel axe shown above. A $10,000.00 reward is offered for return of the axe and information leading to the arrest and conviction of the thief.