The usage of the z-axis is absolutely correct in the article.

The author has essentially defined the x and the y axes. Useful physics (or math), as opposed to what appears to be an arbitrary common convention in whatever field you're familiar with, would suggest placing the z-axis in the direction that is co-perpendicular to the defined x and y axes.

That a z-axis represents height and not depth is the least physics thing I've ever heard. Frankly, I dont think that's a thing at all.