Well there is no such thing as a "single tank" ice course. Even PADI and SDI Ice diving course makes it mandatory to have a redundant air source. This does not need to be doubles and can also be an AL-40 bottle.
I would (personally) still call that a single tank dive. As opposed to diving with doubles. To me, doubles mean back mount, with a manifold, independent back mount, or sidemount. In all 3 configurations, it is intended to breathe from both tanks in normal circumstances. Carrying a pony is for emergencies only and not intended to be used normally - only if there is a problem, which would also prompt termination of the dive. That is one of the primary discriminators between sport and tech diving - the training to deal with issues without going to the surface. A free flow in doubles would mean shutting down a post, switching regs, giving it a few seconds to a minute, then turn that post back on to see if the reg has thawed and is working again. If so, you can continue the dive. I.e. it's a "fixable" failure that does not require (per training) to abort the dive. With only sport diving training, a free flow means the diver is over (per training - unless ice diving training expands your limits in that area).
To me, carrying a pony is an option for any sport dive and is not a technical diving configuration.
So, diving a single tank with a pony still would leave ice diving firmly in the realm of things that seem reasonable for a sport diver (i.e. not a tech diver) to do. It seems logical to me that sport diving certification agencies, like PADI and SDI, would offer a training course for that, rather than force anyone who wants to ice dive to buy doubles and take a 4 (or more) day course. I.e. it seems logical for their to exist a path for people to do the same thing you did.
It also seems reasonable (to me) that a sport diving Ice Diver cert would require limitations like what Cavern does - i.e. tethered, and no more than 200 linear feet from an exit. An Advanced Ice Diver technical cert would require a technical diving configuration (e.g. doubles, redundant computer or bottom timer, redundant buoyancy, etc.) and would allow untethered diving and further distance from the exit (something like Rule of Thirds, with doubles)
Like I said before - so it seems to me, having done neither Cavern nor Ice Diver.
