Turn into the pitch? Must be a confusion of terms.
Faced with a pitch off the plate inside, a right handed batter will tuck his left shoulder toward the catcher. Correct?
I would call that turning away from the pitch.
Reminds me of a kid I had in LL Minors. Big kid, bad attitude; had some potential, but was only there because his parents made him. I was showing him how to turn away from a pitch and wear it on his back so he wouldn't get hurt. He wouldn't hear of it. He said that if he was going to get hit he wanted to get hit in front, not back. Needless to say he never made it past Minors.
maybe a term thing, I always called what you described as turning into the pitch since your turn is into the plate side versus turning out which too me is away from the plate. We both agree that the left shoulder a righty is turned towards the plate side.