Some have stated that kurtosis is the "movement of probability mass from the “shoulders” of a distribution into its center and tails" where "center" is defined as the range between $\mu \pm\sigma$. I was trying to find a proof for this but was unsuccessful. Can anyone point me to a proof of whether (i) larger kurtosis implies greater probability in the $\mu \pm\sigma$ range, or (ii) greater probability in the $\mu \pm\sigma$ range implies larger kurtosis?
I saw a very good reference that pointed out that these intuitive/graphic ideas are not really sound. Kurtosis is often explained as (1) more mass in the tails; (2) less at the center; and (3) some shift relative to the 'shoulders'. I have also seen the claim, as you say, where 'it shifts from the shoulders both in and out, so there is more in both the shoulders and tails.' But all of that is very geometrical, and it raise the question of 'more than what'? And it presumes 'shoulders', which are a vague concept.
I will try to dig up the reference, but I don't think you can prove what you claim. Imagine taking a normal distribution, then take all points within 1-$\sigma$ and put them out at >3-$\sigma$. That will have more kurtosis, I believe, but zero mass in the center.
I will try to dig up that reference, since he did a good job of giving both examples and pointing out that kurtosis is what it is - and while it may have some relation to variance or SD, it needs to be understood (like all moments) for what it is mathematically (which is simply a statistic based on expectations of points raised to a power).