The property called "core-thin" in CatDat does not appear to have been explicitly named many times in the literature. However, the term "essentially gaunt" was proposed by Simon Henry here, and has appeared since in a paper of Jon Sterling. It may be worth considering whether to adopt this name.
(I personally don't have a strong preference either way, because both are descriptive, but I thought it should be brought to your attention in case you were unaware of this terminology.)
I discovered this while creating the nLab page for essentially gaunt category (which the "core-thin" CatDat page could now point to).
The property called "core-thin" in CatDat does not appear to have been explicitly named many times in the literature. However, the term "essentially gaunt" was proposed by Simon Henry here, and has appeared since in a paper of Jon Sterling. It may be worth considering whether to adopt this name.
(I personally don't have a strong preference either way, because both are descriptive, but I thought it should be brought to your attention in case you were unaware of this terminology.)
I discovered this while creating the nLab page for essentially gaunt category (which the "core-thin" CatDat page could now point to).