Types of axonometric projections
I am not familiar how the axonometric projection is dealt in English literature. Here some statements common in German literature (see the German version of axonometric projection):

1) An axonometric projection is a scaled parallel projection (Theorem of Pohlke), mostly oblique (for example: military view, cabinet view), in special cases an orthographic projection, sometimes a scaled orthographic projection (Ingenieur-Axonometrie, standard isometry).
2) An axonometric projection is determined by 5 parameters: the 3 forshortenings vx.vy,vz and the 2 angles alpha, beta between the x- and z-axis and between the y- and z-axis.
3) An axonometric projection is called a) isometric, if vx=vy=vz, b) dimetric, if 2 of the vx,vy,vz are equal and 3) trimetric, if vx,vy,vz are all different. For the popular standard isometry we have besides vx=vy=vz that alpha=beta=120 degree. The standard isometry is a scaled orthographic projection. For the parameters for the military view and the cabinet view see the picture .