There is no contradiction here.

Look at Theorem $2.3$:

Suppose $I\subseteq On$ witnesses $On$ is Ramsey. Then, definably over $\langle V,\in, I\rangle$, there is a transitive class $M$, and an elementary embedding $j :\langle M,\in\rangle\rightarrow\langle V,\in\rangle$ with $j \not= id$.

Note that $I$ is involved as a parameter in the definition of $M$ and $j$. But definability in the Generalizations paper means definability in $\langle V,\in\rangle$ alone.

Contrast this with the sentence

conversely, if $On$ is Ramsey, then such a $j, M$ are definable

from the abstract, where the dependence on $I$ is unstated. In my opinion the abstract is a bit unclear. (That said, this is a great example of why one should always look up the precise statements of the results stated in the abstract before taking them as given.)

There is no contradiction here.

Look at Theorem $2.3$:

Suppose $I\subseteq On$ witnesses $On$ is Ramsey. Then, definably over $\langle V,\in, I\rangle$, there is a transitive class $M$, and an elementary embedding $j :\langle M,\in\rangle\rightarrow\langle V,\in\rangle$ with $j \not= id$.

Note that $I$ is involved as a parameter in the definition of $M$ and $j$. But definability in the Generalizations paper means definability in $\langle V,\in\rangle$ alone.

Contrast this with the sentence

conversely, if $On$ is Ramsey, then such a $j, M$ are definable

from the abstract, where the dependence on $I$ is unstated. "Definability" is being used as a shorthand for "definability from witnesses to the relevant hypotheses." I personally dislike this and in my opinion the abstract is a bit unclear; that said, I understand the impulse to abbreviate results in the abstract, and the corresponding theorem in the body of the paper is clearly stated. I think the takeaway is that results in the abstract or introduction should never be completely trusted (especially when taken at face value they imply something glorious which isn't itself stated).