You also would have seen the one I blatantly pointed out as having a luminous acretion disk, yet no ignition in the star.
Herbig Haro objects, of which DG Tau B is one, have ignited stars in their core. Except Class 0 (there are three classes, defined I believe by their spectral characteristics), which have proto-stars in their cores.
So, all but the very youngest of Herbig Haro objects have ignited stars in their cores (but they haven't neccessarily reached equilibrium yet, like T-Tauri variables).
The literature I have read in relation to DG Tau B suggests that it is
not a Class 0 Herbig Haro.
Proto-planetary disks form
with the star, as part of the accretionary process. In other words, the accretion disk that the star forms within becomes the protoplanetary disk that forms the planets. No proto-star, means no accretionary disk, means no planets.
The difference between a proplyd and a herbig-haro object is that the proplyds are externally illuminated, where the herbig haro objects are not (well, not significantly anyway).
Trolls generally accent something out of context and diminish the real evidence
Yes, they do.