We present nucleosynthesis predictions (HeCNOCl) from asymptotic giant branch (AGB) models, with diffusive overshooting from all the convective borders, in the metallicity range Z⊙/4 < Z < 2 Z⊙. They are compared to recent precise nebular abundances in a sample of Galactic planetary nebulae (PNe) that is divided among double-dust chemistry (DC) and oxygen-dust chemistry (OC) according to the infrared dust features. Unlike the similar subsample of Galactic carbon-dust chemistry PNe recently analysed by us, here the individual abundance errors, the higher metallicity spread, and the uncertain dust types/subtypes in some PNe do not allow a clear determination of the AGB progenitor masses (and formation epochs) for both PNe samples; the comparison is thus more focused on a object-by-object basis. The lowest metallicity OC PNe evolve from low-mass (˜1 M⊙) O-rich AGBs, while the higher metallicity ones (all with uncertain dust classifications) display a chemical pattern similar to the DC PNe. In agreement with recent literature, the DC PNe mostly descend from high-mass (M ≥ 3.5 M⊙) solar/supersolar metallicity AGBs that experience hot bottom burning (HBB), but other formation channels in low-mass AGBs like extra mixing, stellar rotation, binary interaction, or He pre-enrichment cannot be disregarded until more accurate C/O ratios would be obtained. Two objects among the DC PNe show the imprint of advanced CNO processing and deep second dredge-up, suggesting progenitors masses close to the limit to evolve as core collapse supernovae (above 6M⊙). Their actual C/O ratio, if confirmed, indicate contamination from the third dredge-up, rejecting the hypothesis that the chemical composition of such high-metallicity massive AGBs is modified exclusively by HBB.