As far as "which should be used first", I've seen references to gold toners for platinum prints, which implies that gold should be done before platinum. Presumably that applies to palladium as well.
I'm seeing a few vague referencing online about gold toning platinum/palladium prints. So that is possible? If so I think that proves that the information in the book is incorrect, yes?
Both of these posts are correct.
The reason, as one might suspect, is in the details of the redox chemistry.
The propensity to donate or accept electrons is measured by the 'standard electrode potential' which is typically measured in volts.
Here are the values for the metals in question
Au -- 1.5
Pt -- 1.19
Pd -- 0.92
Ag -- 0.80
A metal with a higher potential can be used to replace one with a lower potential.
Thus, gold (Au) can be used to tone prints made with any of the other metals under discussion.
Furthermore, if one gold tones a print already toned with platinum (Pt) or palladium (Pd) the gold will replace the Pt/Pd thereby 'undoing' the first toning.
With split toning the devil is in the details, one needs to pay careful attention to both the concentration of the toners, the time prints stay in the toning bath and the temperature if one wants to have reasonably consistent split toning.
Split toning results will depend on the extent of gold toning since the subsequent toning with Pt or Pd will depend on how much residual silver is present after the gold toning.