Game Development

I don't think that's correct. Here's a drawing I did when trying to get my head around this.

drawing

I find that trying to make sense of terms like "to the left" tricky when we can rotate the directional cube any way we want. For example, in my drawing for "Y-up, left handed", the red X axis is pointed leftwards. However, we could rotate the unit vector cube so that the X axis is pointed right, and the Y axis is pointing up (i.e. the orientation we're most familiar with for 2D graphs). The Z axis would then be pointing away from us, into the plane of the paper/screen.

In contrast, if we oriented the Y-up right-handed cube in the same way, then the Z axis would be oriented as if to come out of the plane of the screen/page, towards us.

These distinctions only matter when we add a third dimension, so the left or right handedness is basically a question of "when we add the third axis to a 2D square made by the other two axes, does the third axis come towards us or away from us? I apologise if this hasn't made things any clearer — I am able to make things make sense by imagining the rotations in my head, but not everyone is able to visualise them like that.