First, I appologise since this should rather be a comment (but I cannot yet comment).
Yet, in the process of reading this question and some of the linked material I got confused, and since I very frequently use the term exponent for abelian groups denoted additively, and know various people who do likewise, I am quite interested in this, too.
I looked at the linked content of Fuchs's book but I failed to locate the part that Mark Sapir is referring to. (I have no access to the book at the moment, and thus would be greatful for a precise quote of the relevant part.)
The only thing I found online is that 'periodic' is used a synonym for 'torsion', but no mention of period of a group.
Also in Lang I was unable to find the mentioned usage, it seems Dylan Moreland made the same experience. Indeed, it seems period of an element is used in Lang also when the notation is multiplicative.
Sorry, for this abuse of 'answer', but I would be grateful for clarification and did not know how to express this except via and 'answer'.