If you consider both curves up to (order-preserving) reparametrization (i.e., the parametrization is a diffeomorphism with strictly positive derivative), then (A) and (B) are equivalent, since any two closed intervals in the Real line are homeomorphic (by an order-preserving homeomorphism). Now, take a homeomorphism $h: [c,d] \rightarrow [a,b]$. Then $\phi':=\phi o h$ is a reparametrization of $\phi$, with $\phi': [a,b] \rightarrow \mathbb R^2$.
So you can always reduce case (B) to case (A), and then, up to reparametrization you can just consider the case (A).