I disagree. The word dexter is Latin. It is related to the Greek word (a cognate, ultimately from Indo-European *dek- see
http://www.etymonline.com/index.php?l=d&p=8 ).
The name Dexter is an Old English surname. It coincides with the Latin word, as in it is spelled the same. If parents DO give the name intending it to mean "skillful", then it is much more likely they have the Latin word in mind.