Not really. Or, rather, not without qualification.slave is a synonym for servant
English law continues to have a law of master and servant. (That's the quaint phrase for employment law.) But it has had no law of slavery for over 150 years.
Servants, in the sense of domestic helpers, were commonplace in the houses of the English middle-class until the First World War, and only disappeared completely after the Second World War. But as a general rule there were no slaves in those houses in the modern era (slavery, as part of English law, generally applied only in certain colonies).
So whether "servant' and "slave" are synonymous depends on context. When I'm told that efreet see others as servants, and knowing also their tendency to twist wordings of agreements, I think of capture, indenture agreements and the like. Not subjecting entire peoples to something like chattel slavery, which is what the notion of "enslaving salamanders" connotes to me.
Now my way of making sense of these various descriptions is only one way, but it's not outrageous. And it's somewhat suggested by the default social backdrop for D&D which, being pseudo-mediaeval but with some of the blemishes removed or disregarded, tends to feature many nobles with their servants, but not so many slaves (except in evil lands). In Greyhawk, for instance, the King of Furyondy is a paladin. Undoubtedly he has servants who help him dress every morning - otherwise it lacks the pseudo-mediaeveal feel - but I'm pretty sure he doesn't keep slaves. This is the sort of context which D&D makes salient to me, and in which "servant" and "slave" are not synonyms.