Reference for “approximately henselian” valued fields

I need some valuation theory in a paper I’m working on. This is not quite within my area of expertise, and I’d like to make the terminology right.

A valued field $(K,v)$ with value group $\Gamma$, valuation ring $O$, and maximal ideal $I$ is henselian if every polynomial $f=\sum_{i=0}^da_ix^i\in O[x]$ with $a_0\in I$ and $a_1=1$ has a root in $I$. (There are other equivalent definitions.)

Let me say that $(K,v)$ is approximately henselian if for every $f$ as above, and every $\gamma\in\Gamma$, there is $a\in I$ such that $v(f(a))>\gamma$. The point is in the following easily shown lemma:

The completion $(\hat K,\hat v)$ is henselian iff $(K,v)$ is approximately henselian.

This surely must be well known, but I couldn’t find it in Engler and Prestel’s “Valued fields”, on which I’m relying as the basic reference.

Question: Does the concept of approximately henselian valuations have a standard name? Can someone point me to a reference for the lemma above?

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I don't know the answers (and I am interested) but note that this property is also equivalent to "for each nonzero ideal $J$, $O/J$ is henselian (as a local ring)". – Laurent Moret-Bailly Aug 14 '13 at 16:01
Thank you, that’s a good point. It got me thinking that “almost henselian” might be another plausible name, and it turns out this is actually used in the literature: jlms.oxfordjournals.org/content/s2-41/1/10.extract , books.google.com/books?id=4NzTiOuirN4C&pg=PA85 . I’ll go with it if there are no other suggestions. – Emil Jeřábek Aug 14 '13 at 16:38