Yes, I mean the completion under Eilenberg–Moore objects, which can be characterised by virtue of being a free completion under a class of weighted limits, e.g. in §4 of Kelly–Schmitt's Notes on enriched categories with colimits of some class.

I'm not sure whether it's relevant to you, but it is possible to characterise the bicategory $\text{EM}(X)$, whose objects and 1-cells are the same as those of $\text{Mnd}(X)$, but whose 2-cells are more general. Depending on your motivation, this might be just as good.

Is this observation regarding the universal property as a triple category of algebras written out anywhere? It seems an insightful construction worth expositing.

I'd certainly be interested to know more general assumptions under which exactness holds, but I don't know of one off-hand.

@JeremyRickard: I don't disagree, but that's the same for every categorical concept. Why are "limit" and "colimit" defined the way they are, rather than the other way around? Because someone made an arbitrary choice at some point, and the convention stuck. It would be better to have symmetric terminology, but it's difficult to change now.

@JeremyRickard: it is a common convention in category theory to dualise the adjectives as well as the nouns, e.g. cofiltered limits and filtered colimits. This is typically useful, rather than being redundant, for precisely the reason mentioned in the question: when considering generalisations, one may need to drop one of the words, but still wants to maintain a distinction between the two concepts.

I would like to clarify that, in my answer, I point out that everything works in the enriched setting, and the examples of enriched algebraic theories you give are also examples of enriched theories in the sense described in my answer (whereas I feel your answer suggests that it is a separate notion). Indeed, quantitative algebraic theories are one of the examples treated in Relative monadicity.

Under the assumptions of 5.4.7, if $\mathcal E$ is an exact category in which regular epimorphisms split, then the category of algebras for any $j$-theory (being the category of algebras for a monad) will also be exact. There is a more general story that remains to be told, where one can work with other notions of congruences (e.g. see Adámek's talk at CT 2024), but it is still a work in progress.

One cannot change history, but I think it would be reasonable to introduce the terminology "coextremal morphism" along with the explanation you have given. "Extremal monomorphism" appears to have been introduced in Isbell's Subobjects, adequacy, completeness and categories of algebras, and the term "extremal epimorphism" in Sonner's Canonical categories. I think it is reasonable to suggest that Sonner dualised the terminology inappropriately.