I have a toilet fitted with dual-flush operated by a large button mounted on the lid, with a smaller button inset. The unit is not cable operated and the only identification is "NF" moulded into the top. Water level is good, no problem there. I can't see anything in the projects, or by searching the forum, which helps.
It's a tall plastic thing with a vertical sliding rod (click-stops) on the back, click-stop adjustable for the lid height setting and an associated curved rod to adjust the amount by which the valve projects at the bottom. There is a click-stopped cover on the front to vary the entry hole for water into the body of the thing.
I've had two or three plumbers trying to get this thing to work, all they seem able to achieve is for it to give a main flush lasting as long as the short one. The latest has been here today, he's replaced it with a new one which is identical. I'm assured that there are no setting-up instructions, which seems very odd.
I stripped the old one to see what could be learned. Not enough! I can see how the short flush works, albeit it only just works as the lever at the back can't be very buoyant. I understand the full-flush mechanism. I cannot decide why it needs an adjustable opening or what effect it might have, and the floats don't appear to leak at all.
It ought to work! Two other toilets in the house are fine, all were replaced after leakage problems. The obvious thing to do would be to investigate one of the others but it all looks so touchy that I'm loth to disturb them!
There are so many of these push button valves now its hard to visualize which one you have ,as a guide the hole in the side with the adjustable opening will vary the amount of water in the flush keep it closed for full flush.
Once the height is set to match the cistern you may then have to fiddle with that other vertical sliding rod to make it work correctly.
They are all a pain ,i also find being a bit generous with the water level can help.
The "NF" logo is something to do with French Regs you will find it on all sorts of things from Towel rails to electrical goods i think its something like our kite mark .
There are better quality valves out there but to change over will mean taking the cistern off to get to the back nut
best of luck .