Is there an established name for cycles $C\subseteq G(V,E)$ with the property that  

$$\lbrace u,v\rbrace\subseteq C\cap V\implies\mathrm{dist}\_{|C}(u,v)\le \mathrm{dist}\_{|G}(u,v)$$ 


I would be tempted to call them *facet*s because vertices and edges that constitute to the boundary of a facet of a polyhedron are prototypical examples of such cycles.



Name for specific cycles in graphs
You are looking for the following:



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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 $u,v\in V(H)$.
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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 $2n$-gonal face $C$ have distance $n$ along $C$, but only distance two in the pyramid (via the apex vertex).


