If I understand correctly, a terminal morphism $(c, \phi)$, is just an edge case of a limit.
So if $(N, \psi: N \to D)$, with the apex $N$, and the base $D$, is a limit, and if $\psi$ is a family of morphisms containing a single morphism $\phi$, and if $c = N$, then their definitions should be equivalent. Is this correct?
And if so, is saying "cone $x$ fulfills the terminal property" a valid statement?