The way I always understood it is that originally the s's (Sanders's dog) style was considered correct, and that the s' (Sanders' dog) style was a shorthand version for informal writing (so in formal writing it would be "sanders's dog" and in letters to a friend it might be "sanders' dog").
But as with all things, informal writing was more popular than formal writing, and eventually the s' style grew in such popularity that it became valid and understood in its own right.
So now effectively we have two "correct" ways of representing the same thing, a "long form" version and a "short form" version. The way I was always taught it (in a stuffy English secondary school) was that consistency within a single text is the most important thing, if you write s's once, you have to use it everywhere else also (ex. quotes).
I think it's much more likely that s' is and has always been incorrect, but the error is so frequently committed that it has become "correct". Language does evolve, after all.
Snark: Given the current rate of apostrophe-abuse, I look forward to the apostrophe merging with the S at the end of words to form a new final-S character, which eventually usurps S entirely re'sulting word's that were 'spelled wrong becoming correct, and creating very weird form's of po's'se's'sive noun's: Jame's''s.
But as with all things, informal writing was more popular than formal writing, and eventually the s' style grew in such popularity that it became valid and understood in its own right.
So now effectively we have two "correct" ways of representing the same thing, a "long form" version and a "short form" version. The way I was always taught it (in a stuffy English secondary school) was that consistency within a single text is the most important thing, if you write s's once, you have to use it everywhere else also (ex. quotes).