As far as I understand one classifies the renormalization group (RG) into the Wilsonian RG and the continuum RG. The Wilsonian RG gives finite predictions by introducing a cutoff $\Lambda$ and absorbs thereby the UV behaviour in the couplings. The flow then describes how the couplings have to change when we change the cutoff. In contrast to that the continuum RG makes things finite by grouping terms together according to a reference scale $\mu$. The flow describes again how the couplings have to change if we change $\mu$ and thereby describes which terms are contributing the most.
I have now 3 questions to the RG:
Is this a correct way to think of these RG‘s?
In which of these categories fits the functional renormalization group? Because it seems that the FRG with it‘s IR cutoff $k$ is intermediate.
Why is the continuum RG called continuum? Is this because in Wilsonian RG one integrates out short distance modes and hence gets a theory on a “lattice“?