Let $R$ be a simply-laced root system in a Euclidean vector space $E$, with inner product normalized so that every root has length $\sqrt{2}$. Let $L \subseteq E$ be the lattice spanned by $R$. Is it true that $R$ is the root system of $L$ (i.e., that $R$ is the set of elements of $L$ of length $\sqrt{2}$)?

This statement is true. There is a more general statement behind it which holds for root systems associated to finite, affine or hyperbolic Kac-Moody Lie algebras, cf. "Infinite-dimensional Lie algebras" by V. Kac, Proposition 5.10(a). This proposition states that you can recover all short real roots by taking elements of the root lattice which have minimal positive length.