To be clear if you get around someone who climbs a bit. When we reference alpine climbing, many times we are discussing big technical routes on big mountains that actually require hard rock and/or ice skills in combination to trodding up steep snow slopes. There is a an alpine grade. In the
Canadian Rockies, if we climb 5000' on a route that is technical in any shape way or form, we tend to label that an alpine climb whether we encounter any snow and/or ice or not. It comes with an alpine and rock grade. For example this one I added last week,
Gargoyle. You would hear us refer to this as an alpine rock route (there typically is no snow or ice on approach). A more classic alpine climb and included in the 50 North American Classics as defined by Roper and Steck would be the
East Ridge of Mount Temple where we carry and use our rock gear and crampons. Same with the
Bugaboos, we cross glaciated terrain in crampons to climb granite spires. All those routes come with an alpine and rock grade. Some even come with an alpine, rock and ice grade.
There are many snow/glacier climbs that most of us view as low grade alpine, they pull an alpine grade, but no corresponding rock or ice grade. This is what you are no doubt mostly referring to as alpine climbs. They are mostly humps with crampons and alpine ax in hand for self arrest purposes. Some of us prefer to solo them, others sew them up with conventional snow and ice protection. The most important advice for such climbs, is either solo them or place adequate pro. Don't ever let anyone suggest that you rope up, but not place "adequate" pro. Depends on the grade of slope, but the majority of the time, you will only be able to self arrest your own fall, and would most likly be incapable of arresting a greater force (your partner's fall which includes velocity).
Waterfall ice climbs many times involve alpine grades as well, i.e.
Polar Circus which not only involves 1000's of feet of hump gain, but has multiple hard and steep waterfall ice pitches.
So an alpine snow route, labeled say Alpine II or III or IV, is completely different than climbing Polar Circus which would carry an Alpine V, WI 5 grade. The later requiring a much higher skill set obviously.
As far as Ouray goes, yes, it is famous first and foremost for its hard and long backcountry ice routes (as the lower 48 goes), the ice park was an afterthought to bring the International comps to town.