I believe you are right, many people don't know much about their roots or the name giving motive of their parents (if they had one besides a pretty sound or someone by the name they knew). (You must admit: Christine is not really a helpful example; it should be Greek, not Latin anyway.)
Now it's exactly the name giving motive of ELITA I've been wondering about. I guess few people would chose the name nowadays knowing that the name has to do with "elite", but maybe some would anyway. So it would be interesting to find out when and where the name first came up.
The characters T and S are often interchangable, but in the case of Elisa/Elita this would suprise me, too.
And don't forget: If one explanation is true, this does not necessarily mean that a different one is wrong. But like you, I would rather stick to the explanation given on this board, unless there was any other evidence.