I suppose you might have some other coordinate system. I even mentioned that in my original post. But that would be extremely anachronistic and out of place on a map like this.
If you took that transverse projection and plotted a graticule on it, you'd get something very wonky looking. If you come up with some other angular coordinate system not aligned to the poles, and then align a tangent equidistant cylindrical projection to it, and the area you are mapping happens to be near the circle of tangency (effectively the equator of the new non-rotation defined coordinate system). Then sure, you could make this sort of work out. This is however such an incredibly bizarre and unlikely thing to do, and so much more complex and unintuitive to explain compared the already hard to grasp concept of map projections in the first place, that I stuck with the simple "equator" rather than trying to explain what a "circle of tangency" was so that I could then use the term to be more precise and cover this insane possibility.