The second one is what JC says on Twitter, but he did a Dragon Talk episode about mounted combat (more recent than the Tweet in question), and he talks there as if the turns are simultaneous. I believe he even uses the term "overlap" to describe how the two turns work.

He talks about it as "the mount's movement becomes your movement", and he describes the tradeoff of having an independent mount as "you have to wait for the mount's turn for it to move.... If it's more important to you that the mount gets its attacks off rather than having all that extra movement occurring on your turn, and that tight coordination, then go ahead, make the mount independent". He believes that the rules for controlled mounts say that you just get to use its movement as your own.