Welcome to the forums.
nifty-stuff.com is where to go for that sort of info. Some of HP's printers use dye-based black and some use pigmented black (as my K5400 does - supposed to be better for text). Who knows, some may use both blacks like some Canon printers do. I'd want to get black that most closely matches the OEM for my model. Some HP refillers were recently all but put out of business by an HP suit for using reverse-engineered, ink that was identical to HP's patented ink. Staples brand refills were apparently one victim of that suit. AFAIK, all the color inks are dye based but there are several formulas for different HP series. HP cartridges (with integrated print heads) should be refillable 3 to 5 times. I know that Canon printers record the switch to clone tanks or refills on models from the last two years or so, but IDK about HP. I got a set of pre-filled, refillable tanks for my HP K5400 - I suppose I'll be finding out soon as my "Yellow Low" LED is on now and you can't mix the clone tanks with OEM, so I'll have to replace all with the clones when the yellow finally gives out...
You can't be prevented from using refill ink, however, if you do, HP can deny service under Warranty by claiming the damage was due to the generic ink. You could try to force them to prove that the damage was really caused by the ink if you have the time and money for legal services. Most won't find the effort worthwhile - after all, one can buy a great, new printer for $150. or less...
.bh.