When a lake listed in the CT fishing guide is listed with the phrase "Use or possession of internal combustion engines is prohibited". Does this apply to gas powered ice augers during ice season?

Hmmmmmmmmm.

It does on Ball Pond. The rule is no motors of any kind including gas or electric ice augers, ATVs, and snow mobiles. It is enforced by the local PD and Encon.

It is a New Fairfield town ordinance for Ball Pond.
That said, these rules issues are usually discussed within the stakeholder groups before they become law. The town(s), lake authority, local politicians, various advocate groups and the DEP (boating division, inland fisheries) all offer an opinion which is negotiated and then an ordinance is offered up to lock the rules in that everyone agrees on. It is rare that something is imposed on a unilateral basis.
So when you go to the Ball Pond launch it says "no motors" period. It may not be what the DEP would do by themselves but it is what the town and the other stakeholders wanted due to some problems that happened in the past and the DEP supported the ordinance because the pond is only 88 acres and there is Squantz Pond and Candlewood within a few miles where you can run most anything you want.
This is why virtually every body of water in the state seems to have its own set of rules and as Blaine says above you need to know what rules are in place for every little spot you fish.
Seems like a hassle but without this dialogue and negotiation local stakeholders would push for total closure of "their" waters. This is how stickers for Candlewood has evolved, basically an effort to limit who uses the lake.

Skip, do you know what category the new propane-powered augers will fall into?

In one sense, it's still a combustion engine. But on the other hand, it burns cleaner, quieter and there is no fuel that can spill into the lake.