I have (had, really, too old for ice now)) climbing axes for steep ice, and a glacier axe, a vintage, hickory handled Chouinard. The Glacier axe is made for walking and had a wide adze head for cutting steps. Longer lengths have become less popular, in part because of the weight, in part because the longer length is actually detrimental to using them in a self-arrest- something that is a very important part of having one. Saved my life once, when I was sidestepping up a steep bergshcrund called the Devil's Chute on the side of Carson Peak in the Sierra. I started sliding , but was able to plant the tip of the axe and jam it in as a brake, using my forearm as a lever. Longer axes can have the tip dig in when your doing this and flip away, or flip you over.

Better you google self- arrest and see it explained.

Then you'd know you lock the handle under your forearm and pull up with the opposite hand, using it like a mechanical brake lever. Of course this puts the upper bodies weight on the pick digging in. Hence the spike tends to tilt towards the ice and a longer handle can jam the spike in. I'm sure there's other methods, but this is the method I learned. I've only had to self-arrest once and hopefully never again. I'm not a practiced mountaineer and hate to give half-azzed advice at something I haven't had to think about in years. You obviously know more than I about the subject, so I'll call this my last word on it. Happy climbing.