I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian."

I'd want to call a local ring $(R,\mathfrak{m})$ pseudo-Henselian if for every [insert appropriate adjective(s)] variety $V$ defined over $R$, if $V$ has a smooth point over $R/\mathfrak{m}$, then that point lifts to a point of $V$ defined over $R$.

This would be in analogy with pseudo-algebraically closed (PAC) fields: a field $K$ is PAC if every geometrically irreducible variety $V/K$ has a $K$-rational point.