There are 2 ideas of symmetry I have seen in Classical Mechanics:
Noetherian symmetry: Here they discuss infinitesimal point transformations where only position coordinates (and their derivatives ) get perturbed. They define symmetry to be those infinitesimal transformations where Lagrangian is not changed. Noether's theorem then relates these symmetric transformations to a conserved quantity $Q = \sum_i p_i f_i(q)$
Hamiltonian symmetry: Here they discuss the idea infinitesimal canonical transformations (ICT), which are actually in (q,p) phase space. It is then shown that the ICT generated by $G$ is a symmetry of Hamiltonian iff $G$ is conserved in time as system evolves by $H$.
I wanted to know if these 2 are equivalent in some sense or not. There are some obvious differences such as the first one is in $q$ space and second in $(q,p)$ space and ICT will allow $G$ (and consequently the transformations) to be dependent on time. But we see that in both cases a translation symmetry leads to a conserved linear momentum and a rotational symmetry leads to a conserved angular momentum.
My query is that is there any formal result which combines the 2 ideas of symmetry?
Specifically is one of these symmetries a subset of the other?