+1 on piston seal lube, although if you're doing this do it properly and remove the piston, clean all surfaces (piston bore, piston sides etc) with some totally non greasy solvent, absolutely NOT brake fluid (which is an oil... ) or white spirit (turps). Meths in a pinch is good, proper brake cleaner (aerosol from motor factors or ToolStation) is best. Lint free cloth for any cleaning, blow out and then solvent allowing it to flash off is best.
If your piston seal is less than 100% perfect you have found your problem and nothing else makes sense without replacing it. (cleaning is required after handling) If working with the piston you should have scrupulously cleaned the faces in the calliper as otherwise you risk accidental contamination of seal damage.
Personally I would use ForkJuice of the products likely to be near a bike for lubing the piston seal. Don't go crazy. A quick psht is all that is needed.
Refit piston carefully- the closer to square when inserted the lower your chance of damaging the seal on insertion.
At this point your system will be voided and contain no fluid so will need a proper bleed.
Grit etc in the threads of the bleed screws at the calliper or master cylinder (lever) will cause trouble by preventing a proper seal.
Poor seals where union barbs enter hoses (eg from shortening) may cause issues.
Imperfect seals in bleed syringes and hose may cause problems. Brake fluid is corrosive and destroys the seals in your bleed syringes and the bleed hoses you're using with them. Nothing you can do about this, they need replacing periodically.
Brake fluid that has not been taken from a brand new, foil sealed, previously unopened container may cause problems- brake fluid absorbs water from the air and there's nothing you can do to stop it and no way you can tell the water's there til your perfectly bled brakes boil unexpectedly causing fade. Your choice.
The Avid syringe method becomes easier with the master cylinder pointing up so the bleed screw is the high point of the system but as its a positive pressure bleed it really shouldn't matter if all the seals in the system and any additional bleeding tools connected are good. Don't know about the procedure on Pinkbike - havent read it - the procedure on Sram's site is correct; there may be many like it.
The alternative method holding calliper above master cylinder and bleeding by expulsion should work reasonably although is likely to be messy and awkward: you should try to close the bleed nipple while fluid is still being injected with this method to avoid ingress at the bleed nipple - assuming that there are no sealing issues. Some cars require this method for bleeding the cooling system as they were designed to be filled under pressure with tools not generally available (TU engined Peugeots in particular)
The positive pressure syringe method should be fine if the tools being used are good and results in minimal mess.
If you suspect trapped air there's no substitute for running fresh fluid through under pressure until you've got it clear.
Long post. Sorry. Good luck!