So the Wikipedia article is a little confusing, but apparently the Roman calendar actually began with the spring equinox in March, so the Ides of March would fall on the 15th by either counting method.

The Julian calendar added two months, but that's where I'm getting confused with the extent of its other reforms, and whether it began with January or March.

However, the central point you are making still stands: "Ides" simply means "the middle," so it would either be the 15th or the 13th of a given month, so "Double Middle" would make no sense in any timekeeping context.