You are entirely correct as to the Latin correlation!
Centuries ago (somewhere around 753 BC according to Roman writers), there was the an early version of the Roman Calendar known as The Calendar of Romulus. (I'm not enough of a history buff to know exactly how widespread the calendar's usage was, but the Wikipedia article I linked to should cover such details)
The Calendar of Romulus consisted of ten months named as follows:
Each month had either 30 or 31 days in it. The problem was, the total number of days only added up to 304 per year; any additional days between December and Martius were not assigned to any month. Time passes, and another king, Numa Pompilius, rectifies this by adding two additional months before Martius.
Side note: If he had appended the two months on the end of December, the last five months' numbering systems would still be valid, but alas, he didn't. Why he didn't, I never learned, but I'm sure there was a good reason at the time.
Over the next centuries, the Roman Calendar gains acceptance across the world while several more minor tweaks to the calendar are applied. But as for the month naming, only two more major changes occur.
There is an urban myth that the months July and August were inserted by the two namesake Caesars into the calendar, causing the following "numbered" months to offset by two; this is not the case.
Quintilis was renamed to Iulius (or Julius, today known as July) in honor of Julius Caesar after his death. Similarly, Sextilis was later renamed to Augustus in honor of a later Roman ruler, Augustus.