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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.

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.

----------------EDIT---------------

I guess I should have made it more clear what I am asking.

a) Does this definition seem appropriate/does it exist already by another name?

b) Is it actually equivalent to Henselian?

c) If not, can you give an example of a local ring that satisfies this property but is not Henselian?

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.

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.