Field
The families
Cylindrical, conic, azimuthal — a taxonomy of construction that says surprisingly little about the properties anyone actually cares about.
Cylinders, cones and planes
The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.
What a standard parallel buys
A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.
The aspect is a free choice
A projection's distortion pattern is fixed relative to its own axis, and where that axis points is entirely up to the cartographer. Rotating it is the cheapest available improvement and it is the one most often left unmade.