This rule always pissed me off because it makes absolutely no sense to me and it's terrible as a shorthand but my physics professors could not stop preaching it for any length of time to come up with something better
If the flow of current follows in the direction the index finger is pointing, then the the magnetic field lines will flow in the direction the middle finger is pointing, the thumb points in the direction that the force is exerted.
Does this help? Short of making sure that your fingers actually are mutually orthogonal. The perspective in OP's picture is all fucked up and also not consistent between the hand and arrows.
"x,y,z" of course is an easy order you learned it in primary school, for it to make sense for the physics you need to... do nothing. Order is arbitrary. The handedness is dictated by the signs of the vectors, I guess if you were to flip the sign of some natural constant the whole thing would suddenly be left-handed.
It of course is also not an explanation of why the stuff is like it is. You say handedness plus assigning things to fingers is terrible as a shorthand? Show me something better, then. Meanwhile, generations of graphics programmers could and can be observed holding out their hand like that and rotating it every which way to make sense of what other people preferring different orientations did. Or keeping track of local vs. global transforms. Gotta know where the wheels are even if the car is flipped over.
(and, yes, for some reason there's no 3d software which uses x as the up/down axis. Whether that means that the industry has a bit of sanity left in it I'm not so sure it might be an accident).
I know what a 3 dimensional axis looks like. Trying to make my cram my hand into a poor approximation of one does not help me personally understand fields.
I don't know how I should be the one to come up with a better shorthand if I'm the one being taught and not understanding.
Maybe it's the resolution of the picture, but those look like finger-teeth to me. I never thought I'd be so upset by molars before, and I've seen those pictures of people with that disorder that makes their entire roof of their mouth nothing but molar mutations, which is pretty fucked up.