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> **Definition.** A subgraph $H\subseteq G$ is called *isometric* if $\mathrm{dist}_H(u,v)=\mathrm{dist}_G(u,v)$ for all $u,v\in V(H)$.

So your cycles could be called **isometric cycles**.

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Note that not all facets of a polyhedron are induced in this sense.
Consider the a $2n$-gonal pyramid for some $n\ge 3$.
Two antipodal vertices of the $n$-gonal face $C$ have distance $n$ along $C$, but only distance two in the pyramid (via the apex vertex).