What the article is referring to is that de Sitter spacetime, which is an idealized universe that only contains vacuum energy, has no initial singularity; it extends infinitely far back in time (and infinitely far forward as well). This is the basic model (at the classical level--see below) used to describe inflation, so in that sense I think the answer is that the "no singularity" result applies to all inflation theories.

That's not quite the end of the story, because the idealized model I just referred to is a classical model; it doesn't include quantum effects. I think the general opinion among physicists is that, if we extrapolate inflation back far enough, we reach a point where quantum effects become important, and the classical model above no longer applies. I don't think anyone expects a quantum model with an initial singularity to replace the classical model in this regime; but at this point we don't know what the correct quantum model is.