"In a model of set theory, this property strengthens V=HOD, but is not first-order expressible"
You may take that as confirmation that "nameable ordinals" (in so far as the modern paradigm would associate that with the notions of definability used in the paper) are not "sharply defined".
The notion of "sharply defined" would correspond with the idea of "definite totalities" that I discussed with you in an earlier post.
Or, comply more closely with the statement itself, note that first-order logic involves certain presuppositions regarding denotation (as noted elsewhere, denotation without instantiation is a notion that may be compared with Russellian description theory). This is stated explicitly in the first sentence of the link: