Kat;
Well, that’s a loaded question. Technically, alpine means above the tree line, so alpine rock climbing is climbing a rocky mountain above the tree line. If you are climbing just rock, the kind of boot/shoe depends on the rock type. Most NW rock climbs can be done in a good flexible pair of cross trainers, though places like a gym, Smith Rock and some granite climbs require bona fide rock climbing shoes (small tight shoes with sticky soles for rock climbing only). I have soloed Mt. Washington here in Oregon (a volcanic rhyolite mtn that is all rock climbing, except for the approach) many times in just my Saucony runners, but I have seen climbers on the routes with rock shoes, too. If you are carrying a heavy pack, though, you need more support than a pair of trail runners. Rock shoes should NEVER be used for anything but rock climbing or you will ruin them.
To me, mountaineering is climbing with technical equipment like crampons/harness/ropes, etc. whether it is rock/snow/ice or a mixture of two or three of those. Some climbers do the approach in a pair of good hiking boots, then don the mountaineering boots (very stiff boots that can be used with crampons) for glacier/ice/snow, though some don’t like lugging the extra wt. of a heavy boot in their pack, so they just wear the mountaineering boots for the hike too (which are very expensive, and very stiff with no flexibility but are warm and great for snow, but hella hard on your feet on dry ground). Ice climbing is just that, climbing ice, whether it be a frozen waterfall, a glacier headwall/crevasse or an icy snowfield, all of which require crampons (the metal spikes that fit on the soles of your boots). You can use a good stiff hiking boot with some crampons for hard snowfields but it is never recommended if you will be on ice or any steep slope (the flexibility of the boot won’t allow you to dig into the ‘good ice’ and will slip off the hold too easily). You may not get to use crampons in your climb school until you are actually on a climb that requires them, but walking in them is easy to master, with practice.
I hope that answered your question. I tried to post this once and lost the connection before I sent it and had to retype…augh!