Long gamma-ray bursts (GRBs) with a plateau phase in their X-ray afterglows obey a 3D relation, between the rest-frame time at the end of the plateau, Ta, its corresponding X-ray luminosity, La, and the peak luminosity in the prompt emission, Lpeak. This 3D relation identifies a GRB fundamental plane whose existence we here confirm. Here we include the most recent GRBs observed by Swift to define a "gold sample" (45 GRBs) and obtain an intrinsic scatter about the plane compatible within 1 with the previous result. We compare GRB categories, such as short GRBs with extended emission (SEE), X-ray flashes, GRBs associated with supernovae, a sample of only long-duration GRBs (132), selected from the total sample by excluding GRBs of the previous categories, and the gold sample, composed by GRBs with light curves with good data coverage and relatively flat plateaus. We find that the relation planes for each of these categories are not statistically different from the gold fundamental plane, with the exception of the SSE, which are hence identified as a physically distinct class. The gold fundamental plane has an intrinsic scatter smaller than any plane derived from the other sample categories. Thus, the distance of any particular GRB category from this plane becomes a key parameter. We computed the several category planes with Ta as a dependent parameter obtaining for each category smaller intrinsic scatters (reaching a reduction of 24% for the long GRBs). The fundamental plane is independent from several prompt and afterglow parameters.