The way I understand it, July (named after Julius) and August (named after Augustus) were added into the calendar year, but placed in the middle. This made September (sept- = 7), October (oct- = 8), November (n-? = 9), and December (dec- = 10) no longer the 7th, 8th, 9th, and 10th months, but the 9th, 10th, 11th, and 12th months.

Not quite. July and August were not months added to the calendar, they were names that replaced the older names of those two months, namely Quintilis and Sextilis. The reason for the displacement of the numbered months by two places is not because of an insertion of months into the middle of the calendar, but because of the extremely early addition of January and February at the beginning of the calendar