Perhaps I misunderstand your question: the nLab has many references besides Fakir. Most of the references on the page idempotent monad discuss the right adjoint (though not always formulated precisely this way).

The theory of categories is not algebraic. Furthermore, if "collection" means set/class, then the collection of categories may be equipped with trivial categorical structure, so it's not a particularly interesting example. As such, I don't see that there's a good motivating example for this question. (However, I haven't downvoted the question.)

Catégories et structures seems the most likely location for a definition, but I can't find a copy.

Here are the links: Catégories avec multiplications and Algèbre élémentaire dans les catégories avec multipliplication. 2-categories appear in neither.

I notice, Part II of Lucyshyn-Wright's Relative symmetric monoidal closed categories I: Autoenrichment and change of base is supposed to contain this result, but it has not yet appeared.

I personally would like to see a precise definition, particularly as we know from other sources that "2-catégorie" was established terminology by 1965.

I should clarify that it is definitely clear that Ehresmann defined double categories in the 1963 paper, which generalise 2-categories, but this is separate from the question of who defined 2-categories explicitly. Both Eilenberg–Kelly and Bénabou use the terminology "2-category", which suggests the meaning is known, but this term does not appear in the 1963 paper as far as I can tell.

Guitart writes that Ehresmann gave the definition of double category in 1961, not of 2-category.

Could you be explicit about where in section 5 the definition of 2-category is given? Probably my French is not good enough, but I only saw what looked like the definition of an n-fold category.

Bénabou in Catégories relatives cites his own "forthcoming" Algèbre élémentaire dans les catégories and Ehresmann's 1963 paper for the definition of 2-category.

In Closed categories, Eilenberg and Kelly cite Ehresmann's 1963 Catégories structurées for the definition of 2-category, though I don't see which definition to which they're referring. However, Wikipedia is citing the 1965 Catégories et structures, of which I'm struggling to find a copy.

@Emily: sadly John Gray died in 2017, so it is no longer possible to ask him about this work directly. (I don't know why his website hasn't been updated accordingly.)