The point is that the cone starts out in $C$, then you test it in $D$. Being a limit cone in $D$ is more than you need by virtue of what you mentioned about some test objects in $D$ not being visible to $F$.
Suppose you had a cone $\ell$ in $C$ whose image under $F$ is a limit cone in $D$. Let $c$ be another cone in $C$ for the same diagram, then $Fc$ will be a cone in $D$ and so by the universal property (of $F\ell$ in $D$) this admits a unique morphism $Fc\to F\ell$. Now, since $F$ is fully faithful, this morphism arises as a unique morphism $c\to\ell$ in $C$, proving that $\ell$ has the universal property of being a limit cone in $C$ as well.
I suspect you might be confusing this with the property of preserving limits. If a fully faithful functor cannot see all the objects of $D$ (i.e., is not essentially surjective) then the fact that there are test objects in $D$ that $F$ cannot see will make it possible that a limit cone in $C$ will no longer be a limit cone in $D$ (for instance, $\{0\}$ embeds fully faithfully into $\{0\to1\}$, but $0$ is only a terminal object in the former).