After looking for a long time for an answer to this same question, I found a couple interesting links https://www.jstor.org/stable/2277400?seq=1#page_scan_tab_contents

where we can only see the first page but that's where the derivation is. The "standard deviation by dr Sheppard" is given by something called the Asymptotic distribution of moments, of which you can see a bit here

https://books.google.com/books?id=Uc9C90KKW_UC&pg=PA126&lpg=PA126&dq=Mst+pearson+Sheppard&source=bl&ots=Kvw0xTLzps&sig=pyHVB_ybjsnb_0QOBDHST6SRi-M&hl=en&sa=X&ved=0ahUKEwimjvjQ8NnSAhWEppQKHRqbC1sQ6AEIIjAD#v=onepage&q=Mst%20pearson%20Sheppard&f=false

The reason for the "n-2" instead of "n" in the root, is that your formula assumes a t-distribution with n-2 degrees of freedom, while the one in the links assumes a normal distribution.

After looking for a long time for an answer to this same question, I found a couple interesting links:

$\bullet$ The Standard Deviation of the Correlation Coefficient, where we can only see the first page but that's where the derivation is. The "standard deviation by dr Sheppard" is given by something called the Asymptotic distribution of moments, of which you can see a bit in the following source.

$\bullet$ A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713-1935.

The reason for the "n-2" instead of "n" in the root, is that your formula assumes a t-distribution with n-2 degrees of freedom, while the one in the links assumes a normal distribution.