Yes, it seems that limes came first, though table 2 of the last link below indicates that lemons have been as food longer than limes.
Citron medica is the fruit species from which both originated and as you indicated, the diversity of lime species makes it older than the lemon.
I was hoping that somebody had a copy of a (very old) National Geographic in which they featured "Fruits of the World". There, I read that limes were definitely older than lemons and that in fact, lemons were hybridized from limes. It was probably from before 1950, though I bought my copy recently at a library book store and has since disappeared. That feature on fruits was so extensive that it was covered on two consecutive issues of the magazine.