Klammerreflex
I searched that originally and only got German things, so I figured we just hand another term for it and just googled "baby grip finger reflex" and that got me the term we use.
Whoops! I realize I linked the wrong wiki! My quote was from your Moro reflex link.
I like when I remember to check other countries' wiki pages, I always forget they are all different since they're community made and not just software translations.