A few answers from someone who owns a Variax 500 since 2003...
-IME, "LP sound or other warm guitar models" DO quack: it's one of the prominent features of PAF pickups and one of the reasons why vintage Gibson are considered as unique sounding VS modern ones. I've tried many PAF replicas and only a few of them are able to reproduce this tonal feature, BTW.
-That said, yes, the Variax quack exists and is not exactly the same than a "vintage PAF quack". On a frequency chart, it appears as a narrow peak around 1khz, typical of piezo transducers...Personally, I've solved it by stoping to use modeled sound with overdrive (I've mounted mag pickups in my Variax for that and I use the Variax models with clean tones only).