how to adjust the derailer - apart from the L/H limiting screws Pardon my newb terminology. My bike's chain will not stay on the H gear (the big ring) The front derailleur will not allow it.



I have loosened the limiter screw out to it's maximum and, still, no dice. Clearly there is more room to move, as I demonstrate manually:



I want to know what to attempt to adjust next. Clearly there is a bias toward the Low cog. Ideally I'd want to tighten that limiting screw so I'd have more room to manouvre in future.

Which of the other hinges should I try to work with next?



Every video I've found online so far only deals with those limiting screws.
You need to loosen the screw that holds the cable and pull that cable out a little bit.

The limiter screws that you've used so far are rather used to prevent the chain from falling off, and it's a finer tuning than you appear to need now. Thus it would be useful to level them out before pulling the cable. After you adjust the cable, take a few minutes to make sure the derailleur does not throw the chain over the outer gear.