What the meaning of regen pet and tagged pet ?

Regen pet: post 3 of this guide covers regen pets in as much detail as is likely needed
Tagged pet: post 9 of this guide is a glossary that includes words like Tagged

but in short, regennable pets are pets of species that went through a revamp and the species had traits shuffled or split, or some other significant change. Pets with traits/colours affected have all their traits removed and reverted into trait points that can be reassigned, as their trait setup is no longer viable.

Tagged pets are first-gen themed pets- either tamed or exclusive-, 'Tagged' referring to the tag in their genetics: