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.

There is no better one. Your hand is the only obvious, also, convenient, piece of equipment you always have at the ready that is capable of clearly and intuitively representing three mutually orthogonal vectors. They also happen to be quite freely orientable in space. There are exactly two distinct ways axes can be constructed in 3d, and your left and right hand naturally show the one or the other, try it: While making fancy fingerguns you can never have all fingers be parallel, if two pairs are parallel then the third is antiparallel. You could use fancy terms like chirality but that literally means handedness.

And if you had tried to come up with something better you'd probably have realised all that.