Conformally invariant theory. Relationship between conformal transformations and conformal rescaling (Weyl scaling)

So, I’m learning about Twistors, and in every book I’ve read they say the same:

“If a flat theory is Poincaré-invariant and it is invariant under conformal rescaling (Weyl scaling), it is then conformally invariant (in the sence of conformal transformations)”

First let me say that I can’t find a proof about this, but for me this looks like another way of stating the Liouville Theorem (or maybe may be it’s a consequence).

Second, Is this sentence the reason for study only conformal rescaling (in the flat space)?