Sorry, you're right: I swapped 1 and 2 by mistake. Dubuc proves 2 and 3 in Theorem III: the equations (1) and (2) state that the equivalence commutes with taking categories of algebras. I believe that the monads are equivalent follows because taking the codensity monad of a functor means the monad structure is uniquely determined, so it suffices just to check that the left adjoint is correct. (I think there are more elegant proofs of this correspondence, but there's often a trade-off between a classical proof and an elegant one!)

@MartinBrandenburg: I shall try to do so a little later.

My understanding is that the unenriched correspondences were part of the folklore after Linton, so all that was left was the enriched versions. I agree it'd be nice to see the intermediate step, but Dubuc's paper is dated 1970, just one year after Linton's paper, so it's unlikely such a reference exists.

In my copy of the book, 1 is Theorem A.38 (Theorem A.42 for multisorted); and 3 is Theorem A.21 (Theorem A.41 for multisorted). (Perhaps there are different editions.)

@PaulTaylor: I should have thought to look there, thanks! I shall add a reference imminently.

@TimCampion: that's true. I had been thinking that there was a strong connection between the injectivity characterisation and the Kan injectivity characterisation, which would justify interchanging them, but now that I think about it, it's not so clear. It would be good to have something like Proposition 4.4 of Escardó's Injective spaces via the filter monad, which could justify Kan-injectivity in this setting.

@TimCampion: ah yes, I took the liberty of choosing a stronger definition of extension than in your question – it seemed like the natural generalisation from the posetal case considering the relationship between LKEs and cocompleteness! I'm not sure I knew of this property regarding left extensions along $I \to I^\rhd$, which does greatly simplify this answer – but I think in any case it's more satisfying to have a more conceptual understanding that generalises more easily :)

After posting this, I started searching around and found Kan injectivity in order-enriched categories by Adámek–Sousa–Velebil, where they study objects in order-enriched categories that are "Kan-injective" with respect to a class of morphisms, and show them to be characterised as cocomplete objects. Presumably their development could be carried out at the same level of generality as in this answer to give a more general understanding of Kan-injectivity.