What does the probability subscript mean in Shannon's secrecy definition?
This is standard notation in information theory but it is redundant as given here, usually it is used as below. 

For example
$P_M(m)$ would be used instead of $P_M(M=m)$ both of which refer to the random variable $M$ and the probability that it equals $m$. 

$P_{M,X}(M=m,X=x)=P_{M,X}(m,x)$ would refer to a joint distribution in the same way.