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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.
Note that not all facets of a polyhedron are induced in this sense. Consider the a $2n$-gonal pyramid for some $n\ge 6$. Two antipodal vertices of the $n$-gonal face $C$ have distance $n$ along $C$, but only distance two in the pyramid.