Note that it is a DistanceFunction option for ImageDistance. Maybe that helps some to see the context where it is used.

The relation to SquaredEuclideanDistance is:

NormalizedSquaredEuclideanDistance[x, y] ==
   (1/2) SquaredEuclideanDistance[x - Mean[x], y - Mean[y]]/
      (Norm[x - Mean[x]]^2 + Norm[y - Mean[y]]^2) 

So we see it is "normalized" "squared euclidean distance" between the "difference of each vector with its mean". I guess that was too long for a function name.. In any case the note under properties and relations ".. includes a squared Euclidean distance scaled by norms" makes little sense.

The 1/2 is just there such that the answer is bounded between 0 and 1, rather than 0 and 2.

This is also asked elsewhere:

https://stats.stackexchange.com/questions/136232/definition-of-normalized-euclidean-distance

But I don't think an answer exists anywhere. I spent a fair amount of time looking for it myself. Ideally a function like this should directly point to a commonly known function that I can read about somewhere.

I've forwarded a report to the developer of this asking that clarification be added to the documentation about this function.