Definition of an $\omega$-huge cardinal

Question

Martin and Steel's paper "A proof of projective determinacy" defines an $\omega$-huge cardinal to be an I2 cardinal but more recently you see it being defined to be an I1 cardinal. Is there a standard definition?

Answer 1

There are three definitions. The first (That I use) is that $\kappa$ is $n-$huge for every $n$. In "Large Cardinals beyond Choice" they are used to refer to $I3$ cardinals. I have seen them commonly defined as $I2$ cardinals. I think it is uncommon to have them be used as $I1$ cardinals. I have seen them be used though, for critical points of elementary embeddings $j: V_{\lambda+1}\prec V_{\lambda+1}$ with $V_\kappa\prec V_\lambda\prec V_\gamma$ for some $\gamma\gt\lambda\gt\kappa$.