When comparing the energy of the LDA0 and SVWN (Slater exchange and VWN(1980) correlation), I get different results.
So what is the LDA0 functional in Psi4? I can't really find literature about it.
As Tyberius already stated in a comment above, LDA0 is a hybrid functional. However, you can't even expect two different LDAs to give the same results! SVWN5 is different from SVWN3 is different from SPW92.
The functionals available in Psi4 are documented in the manual. Psi4 does not implement any of the functionals itself, but relies on Libxc for the implementations. The available functionals in Libxc are listed here.