I know you wanted to avoid this topic, but you seem knowledgable about it and I can't help but ask! But is there anything in your opinion that makes one cold water reg better than the other? Or is there anything on a reg that's needed uniquely to ice diving?
I'm sure there's lot of other threads on ScubaBoard that discuss this but I'll give you my quickie take on it all...
The number one thing is having a 1st stage that's environmentally-sealed. You don't want cold water to have access to any of the colder innards. IMO, in silty water like we have here in Colorado, an environmentally-sealed reg is a good idea regardless of temperature.
The second stages are particularly prone to freeze-up. And yeah, your breath has moisture that will cause them to freeze and free-flow.
Beyond that you get into areas where reasonable divers may disagree. Many manufacturers make "cold water" regs, but that doesn't mean they won't freeze. Thankfully, the failure mode is almost always free-flow, which is alarming but not nearly so much as having one fail to deliver air entirely.
One more big note about ice diving...
PatW is right that there are serious risks. However, the reality of recreational ice diving
classes is that safety comes first, and in some respects it may actually be safer than a typical open-water dive. Let me explain...
1. Many classes are 'light weight'. By that I mean they are supposed to be a fun introduction to ice diving. A competent instructor will not let you get into trouble.
2. You'll probably never go ice diving without an entire team of people planning and participating. The logistics are immense.
3. You never lose sight of the hole. There should be an emergency exit too.
4. You're tethered, and in constant communication (through the tether) with the surface tenders. In an emergency they will literally drag you out.
5. If the tether doesn't work, there is an emergency diver standing-by on the surface ready to rescue you.
To make the most of it, both in terms of comfort and safety, I'd recommend becoming experienced with a drysuit first -- which is a challenge in itself. A lot of people say "Burrrrr!" about ice diving, that only means that either a) they haven't tried it, or b) they have tried it but were wearing insufficient protection. Your lips and cheeks will definitely get cold, but if you're wearing the right protection -- and this is true for any kind of diving -- then you should be fairly comfortable.
Stay warm!