By "regular" Eisenstein I mean:
Let $R$ be a UFD with field of fractions $F$. If $f(x)=a_0+\cdots+a_nx^n\in R[x]$ and there exists a prime $p\in R$ such that $p|a_i$ for $0\le i\le n-1$, $p\not\mid a_n$, and $p^2\not\mid a_0$, then $f$ is irreducible in $F[x]$.
We also have the generalized Eisenstein:
Let $D$ be an integral domain with field of fractions $K$. If $f(x)=a_0+\cdots+a_nx^n\in D[x]$ and there exists a prime ideal $P$ such that $a_i\in P$ for $0\le i\le n-1$, $a_n\notin P$, and $a_0\notin P^2$, then $f$ is irreducible in $D[x]$.
Now, in the presence of Gauss's Lemma (which holds for GCD domains), the latter version implies $f$ is irreducible in $K[x]$. Taking $P$ to be $(p)$ for some prime element $p$, we recover the regular version.
It seems to me that we can amend the regular statement to replace "UFD" with "GCD domain." For some reason, I can't find this version anywhere else. I just wanted to check that I'm not missing something, especially since there is some subtlety with prime ideals / elements when going back and forth between UFDs and GCD domains.
Bonus question: is this even useful? The only example of a GCD domain which is not a UFD that I know of is the algebraic integers. However, there are no primes at all in AI since, for example, $x=\sqrt x\sqrt x$. Is there a reason we talk mostly about UFDs even when the same thing applies to GCD domains?