How do look the categories equivalent to models of finitary and infinitary limit sketches, respectively ?
There are at least $2$ interpretations of the word finite here: the cardinality of the set of cones in the sketch is finite or the domains of the cones are finite; which is the better understanding ?
It is not obvious to me whether by allowing the word infinitary in our chosen meaning creates less or more categories ? I have some idea, but I would not bet on it.
A category is equivalent to the category of models of a limit sketch if and only if it is locally presentable. This is Corollary 1.52 of Adámek–Rosický's Locally presentable and accessible categories. This restricts to locally $\lambda$-presentable categories and limit sketches with $\lambda$-small diagrams, as noted in the remark following Corollary 1.52 (where $\lambda$ is a regular cardinal). If $\lambda \trianglelefteq \mu$, then locally $\lambda$-presentable categories are locally $\mu$-presentable (Theorem 2.11 ibid.), so in this sense, locally $\mu$-presentable categories are more general.