Snaffle has a concrete, agreed-upon definition: It is a shankless bit with no leverage. All other variables are up in the air. This is not the case with the TT, which has an ambiguous definition from the outset (we're already looking at three different countries with three different TT definitions, and when it comes to your "true" American Tom Thumb, even the straight-shanked version often have the tiniest bit of curvature....so are they not TT's, either?).
Now, in common Western terminology, it is gradually becoming more and more acceptable to refer to a single-jointed mouth as a "snaffle mouth," knowing full well that such a bit is not a true snaffle, but meaning to shorten the terminology of the mouthpiece from "single jointed and broken." Hence your Argentine snaffle, etc., which of course is not a true snaffle, but which comes with the attached moniker regardless. The "snaffle" debate has been hashed out even by big-name "cowboy" trainers in mainstream equine magazines.