I think I'm in the process of understanding something very subtle here, and I could use an expert's double check. So basically, my question is whether what I write is correct.
(Non-finitary) GATs, EATs, and limit sketches all correspond, in the sense that a category is a category of models of a GAT iff it is one of an EAT iff it is one of a limit sketch (iff it is locally presentable) - and converting the theories is not even that hard.
Sites are a special case of the above in the sense that every sheaf category (Grothendieck topos) over some site is in particular locally presentable, and therefore a model of some GAT, EAT, and finite limit sketch. Once more, converting a site into a GAT / EAT / limit sketch is not very hard (at least if your underlying category has the necessary pullbacks).
What I was a bit surprised about is that we cannot convert back. I now think the reason for this is that there is a subtle difference between limit sketches and sites (and the definition of their models/sheaves), namely that in a limit sketch, you get to choose cones, which go from a diagram to a tip and thus allow you to define what compatibility of morphisms is required before they can be pasted together to obtain a morphism to the limit (which is the image of the cone's tip in a model), whereas in a site, you have to work with covering families which go from a bunch of loose objects to a tip, and the compatibility criterion in the sheaf condition is uncustomizable.
