There is a tension between (1) interpreting proper class talk in set theory as talk about first-order formulas and satisfaction; and (2) taking it to be an interesting and non-trivial result that there is no (non-trivial) elementary embedding from V into V and/or taking it to be an open question whether there can be such an e.e. in the absence of choice. Basically, there is a very simple proof that there can be no definable e.e. from V into V (see Suzuki (1999)).
This tension was recently highlighted by Hamkins, Kirmayer, and Perlmutter (2012) (and pointed out here and here). There, the resolution was to give up on (1), since accepting it "does not convey the full power of the [Kunen's] theorem" (p. 1873). But this is perhaps the only place I've seen this issue addressed. For instance, Kanamori seems to hold both (1) and (2) in The Higher Infinite: "By “class” in the ZFC context is meant definable class,... $x \in M$ is merely [a] facon de parler" (p. 33); and "[t]he following unresolved question [i.e. whether there could be an e.e. from V into V in the absence of choice] is therefore of foundational interest" (p. 324).
My question is: how do other set theorists prefer to resolve this tension?
Hamkins, J., Kirmayer, G., Perlmutter, N. (2012) ``Generalizations of the Kunen inconsistency". Annals of Pure and Applied Logic, 163, 1872–1890.
Suzuki, A. (1999) No elementary embedding from V into V is definable from parameters. Journal of Symbolic Logic 64, 1591-1594.