Take a hash function $H$.
There will be pairs of messages of the same length $M,M'$ for which $M\ne M',H(M)=H(M')$.
Let $P(x)$ be a function counts the number of $1$s in $x$. $d=P(M\oplus M')$, or in words, $d$ is the number of bits where $M$ and $M'$ differ.
Let $T(H)$ be the minimum value of $d$ for $H$. This is the metric. It does not say anything about how hard it is to actually find these colliding messages, only that a certain minimum number of bits must change to produce a collision.
Is this metric actually useful? Is it okay for $T(H)$ to be small, as long as actually finding such a collision is infeasible, or does a small $T(H)$ imply $H$ is weak?
Are there bounds on $T(H)$ for any of the popular hash functions? (SHA2 family, SHA3 family, BLAKE2 and its variants)