Though Linton's An outline of functorial semantics does contain the essence of the results and proofs of the monad–theory correspondence, it is true that the terminology and style make it difficult to extract the results as we would expect to see them today.
The earliest reference of which I am aware in which the monad–theory correspondence appears in the modern form is Theorem III of Dubac's Enriched semantics–structure (meta) adjointness. Dubac establishes an equivalence between (large) $\mathcal V$-theories and $\mathcal V$-monads on $\mathcal V$, which in particular implies the classical result.
For the finitary version, the result (again in the enriched setting) appears as Theorem 4.3 of Power's Enriched Lawvere theories.
As far as I am aware, the only paper in which both results follow from the development therein is Lucyshyn-Wright's Enriched algebraic theories and monads for a system of arities: in particular both the finitary and the large monad–theory correspondences follow from Theorem 11.8.