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    $\begingroup$ Upon reflection, I think these characterizations don't say much more than "$T-Alg$ is coextensive iff the full subcategory $(T-Alg)_{fp}$ of finitely-presentable objects is coextensive", which seems to beg the question a bit. But at least this is something in the literature. There are a few comments in the paper which give more information. For instance, $T$ is Morita equivalent to a single-sorted theory with at least one constant. $\endgroup$ Commented Dec 28, 2021 at 13:16
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    $\begingroup$ Why does ${(T-Alg)_{fp}}$ have finite products in that case? $\endgroup$ Commented Jan 18, 2022 at 20:57