Cantor's Attic defined tall cardinals as follows:
A cardinal $\kappa$ is $\theta$-tall iff there is an elementary embedding $j:V \to M$ into a transitive class $M$ with critical point $\kappa$ such that $j(\kappa) \gt \theta$ and $M^\kappa \subset M$. $\kappa$ is tall iff it is $\theta$-tall for every $\theta$; i.e. $j(\kappa)$ can be made arbitrarily large.
It also claimed that every strong cardinal is tall. Why is that? It doesn't follow immediately from the definition of strong cardinals, and as far as I know, extenders can't even ensure that $M^\omega \subset M$
Most of what I know about extenders comes from the paper Double helix in large large cardinals and iteration of elementary embeddings by Kentaro Sato, whose definition 3.2(3) (which relies on definition 3.1) seems to be roughly the same as what other set theories call extenders.