These two terms are quite rigorous in the way they're defined and used:
Fermi correlation: This is the correlation found in the electron exchange term in a Hartree-Fock calculation, describes the correlation between electrons with parallel spins, and prevents two parallel-spin electrons from being found at the same location.
Coulomb correlation: This is the correlation between any two electrons regardless of whether the spins are parallel or not, since any two charged particles will experience a Coulomb repulsion based on the distance between them.
These two terms are often used more loosely:
Static correlation: This is the correlation associated with a mean-field-type SCF (self-consistent-field) calculation. For example, HF-SCF or MCSCF.
Dynamic correlation: This is correlation associated with the "movement" of electrons, and is usually associated with post-SCF calculations such as configuration-interaction or coupled-cluster calculations.
It is usually said that when doing a Hartree-Fock SCF calculation, you only get static correlation, and therefore all of the Fermi correlation from the Hartree-Fock SCF is associated with static correlation.
However, if you're doing an MCSCF calculation, it is still said that you are getting static correlation, but it would seem that there is much more to this static correlation than what you would get from a Hartree-Fock SCF calculation, and the Fermi correlation would seem to constitute a much smaller percentage of the overall static correlation.
The terms "static" and "dynamic" correlation can be confusing though, because in an MCSCF calculation, for example a CASSCF calculation, you're doing FCI in the CAS at every step of the SCF, and I mentioned earlier that configuration interaction gives you dynamic correlation, but often (for example here) it is said that MCSCF does not give dynamic correlation. Furthermore, there's SCF-type methods such as pCCD (pair coupled cluster doubles), which is also known by some as AP1roG that recover a significant amount of correlation (in the case of pCCD/AP1roG, the energies obtained can match those of DOCI, or doubly-occupied configuration interaction, which is FCI but with all spatial orbitals always doubly occupied in all considered interacting configurations, to within 1 micro-hartree of the exact DOCI energy). When something like pCCD/AP1roG agrees with DOCI up to the micro-hartree digit, it must include dynamic correlation, but I'm not sure where to draw the line between static and dynamic correlation in this case because we don't have something like a Hartree-Fock reference.