Hello Virginia,
I've downloaded all of the scripts in part to make answering questions such as this a breeze. A quick Windows search through the files reveals the following: The name 'Alexander' is mentioned in two scripts. In the pilot, Lex owns and discusses the history of Alexander the Great's sword. Although it could be argued that part of his interest in this artifact is that it belonged to his namesake, such an argument would be purely speculative. The other time it is used, in Vatman, is as a surname.
So as far as official L&C canon is concerned, 'Lex' is not necessarily short for 'Alexander' in Luthor's case.
Joy,
Lynn