Question about computable function with recursive domain

Question

Assume there is an $x\in\mathbb{N}:dom(φ_{M_x})\in$ REC. Obviously $φ_{M_x}$ is computable and since $dom(φ_{M_x})$ can always decide if something belongs in its domain or not, doesn't that mean that $M_x$ corresponds to a decidable language? If that's true, what's the difference between saying that $φ_{M_x}\in$ REC?