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

I'd want to call a DVR $(R,\mathfrak{m})$ pseudo-Henselian if for every [insert appropriate adjective(s)?] variety $V$ defined over $R$, whose generic fiber is absolutely irreducible, say, then 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.