I understand that given a root system $\Phi$, by Serre's theorem there exists a Lie algebra $L$ with root system $\Phi$. Also isomorphism theorem implies that any two such $L$ are isomorphic. That means $\Phi$ determines $L$ up to isomorphism.
Now given $L$, we can find two maximal toral subalgebras $H$ and $H'$, therefore two root systems $\Phi$ and $\Phi'$. Humpreys says in his book
In order to show that $L$ alone determines $\Phi$, it would surely suffice to prove that all maximal toral subalgebras of $L$ are conjugate under Aut $L$.
I want clarification on two things
1) By saying $L$ alone determines $\Phi$, does he mean $\Phi\cong \Phi'$ or stronger $\Phi = \Phi'$, that is, does $L$ determine its root space completely or up to isomorphism?
2) I don't see how is that by showing all $H$ are conjugate under Aut $L$ will imply the result in 1).
1) He means up to isomorphism.
2) If any two $H$ are conjugate by an element of $L$, then this conjugation will induces an isomorphism between the corresponding character lattices, which will further induce an isomorphism between the corresponding roots systems.