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@MartinBrandenburg: I was looking at the paper again today, and I realised I had overlooked Lemma 10.2, which does establish a relationship between the categories of monads and "theories" (for a suitable notion of theory), and in particular monad morphisms and morphisms of "theories". In this light, I think it is appropriate to attribute the full equivalence to Linton; however, Dubuc's paper is certainly the first that contains the modern monad–theory correspondence (with the modern definition of "theory"). I've updated my answer accordingly. Apologies for the confusion.
@Adrien: I think I misunderstood what you meant initially. But in this case, say that $B$ has only some finite colimits, and take $\iota$ to be the identity. Then $B_A = B$, but this doesn't have all finite colimits, so is not correct. By "the class of colimits in the image of $\iota$", I mean the diagrams with colimits in $B$ that arise by postcomposing diagrams with colimits in $A$ by $\iota$.
@Adrien: if I'm not misunderstanding your description, I think it is not correct. For instance, take $\iota$ to be the identity on $B$. Then the finite colimits of representables of $B$ form the cocompletion under finite colimits, which ignores all existing finite colimits in $B$. In answer to your second question, you ought not to need any requirement on $\iota$, since $\iota$ itself is not crucial: the only important data is which colimits are in the image of $\iota$.
I emailed Dubuc, who believed he was the first to prove the full equivalence between monads and theories (rather than just the bijection between monads and theories as Linton does), so I think it worth citing both authors for the origin of the modern correspondence.
