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).