As pointed out by several people already, some information can be found elsewhere, as in here. And also the differentiation between DFT (exact) and density functional approximations (DFAs), as pointed out regularly by Mel Levy, can be found there.
However, I think there is one aspect missing, and here I would like to quote my late PhD supervisor Jaap Snijders. The most important aspect to know if a method is ab initio or not, is related to the integrals. If the integrals can be computed from the beginning, the method is ab initio; if not, then not. In DFT, DFAs and wavefunction methods, the integrals can be computed, and hence, these methods are ab initio. In semi-empirical methods (AM1, PM3, DFTB, xtb), some of the integrals are either estimated or approximated (from e.g. DFA results in case of DFTB/xtb), and therefore, these methods are not ab initio. Likewise for e.g. the Empirical Valence Bond method, which like the name already indicates, is empirical.
Whether or not a method gives the exact energy is a different aspect. In that case only Full CI with infinite basis set and DFT give the exact energy, all other methods are approximations. By choosing a basis set of a certain size, one is approximating; by using "only" CCSD(T), one is approximating; by using a density functional like PBE, B3LYP or r2SCAN, one is approximating; etc.