[Rockingham]

Correct, My understanding is that the term slave came from the Latin Sclāvus (“Slav”), because Slavs were so often forced into slavery. The Latin term is said to be rooted in Byzantine Greek usage.

[Verginius Rufus]

The term "Slav" comes from the Slavic languages. Slava in Russian means "glory" or "fame" (cognate with Ancient Greek kleos--the Slavic languages are "satem" languages which have an S where the "centum" languages like Latin, Greek, and the Germanic languages have a K [which became an H in the Germanic languages--so English "hundred" is cognate with Latin "centum"]).

Russian slovo means "word."

Among Slavic groups you get ethnic names or place names like Slovak/Slovakia, Slovene/Slovenia, Slavonia, etc. Either people thinking of themselves as the glorious ones or the ones who speak intelligibly (unlike the Vlachs and Germans and other non-Slavs).

The Latin term would be formed from the word the Slavs used for themselves.