I can’t claim to understand the intricacies of the gear, but I can say when going through the DI alone on my SVT-CL, the V4B, and the V12…that sound is present on all of them. Same with the SGT-DI. Even without the SGT circuit turned on.
I don’t have a Fridge or the 212AV to compare with, but I still hear that kind of sound through my Eden as well as the VB212. For what it’s worth, both the SVT and V4B sound great through the VB212. I do think the cabinets contribute to a difference in sound, but for example I’ve also tried the Mesa D800+ and the Aguilar TH700 through the DI and my Eden and they were very different (not in a bad way at all, they’re excellent amps). So in that way, I think the head does have a certain sound of its own. Not sure if that helps.