I have seen the concept of "characterizable" used for both cardinals and ordinals. It appears to stem from the Ph.D. dissertation of S.J. Garland (1967) part of which can be found in GARLAND, S. J. (1974),  [Second-order cardinal characterizability][1] and has been reused in e.g. George, B.R. (2006) [Second-Order Characterizable Cardinals and Ordinals][2] and Kunen (1971), [Indescribability and the continuum][3].

It boils down to the following definition, with some minor variations: A structure $M$ is (second-order) *characterizable* if there is a formula $\varphi$ such that $M\models \varphi$ and all model of $\varphi$ are isomorphic to $M$. From there, an ordinal $\alpha$ is deemed characterizable if the structure $(\alpha, \in)$ is characterizable and it seems natural to extend it to either $L_\alpha$ or $V_\alpha$.

This is very close to what JD Hamkins suggested but the name maybe has the advantage of being more straightforward.

And another similar notion can be found in Barwise (p.257): An admissible set $A$ is *self-definable* if there is a theory $T$ such that some extension of $A$ into a structure for $T$ is a model for $T$ and any extension of $B$ being model of $T$ implies that $A \simeq B$.


  [1]: https://people.csail.mit.edu/garland/publications/Reprints/1974-Second_Order_Cardinal_Characterizability.pdf
  [2]: https://link.springer.com/article/10.1007/s11225-006-9016-7
  [3]: https://books.google.fr/books?hl=fr&lr=&id=TVi2AwAAQBAJ&oi=fnd&pg=PA199&dq=Indescribability%20and%20the%20continuum%20K%20Kunen&ots=LYbb8YOhR2&sig=yq8_ReERiGim_XNvnnC6jVT6U1Q&redir_esc=y#v=onepage&q=Indescribability%20and%20the%20continuum%20K%20Kunen&f=false