What is potential energy in special relativity?

Special relativity doesn't alter the fact that interactions between particles "store energy" in the form of "potential energy," alhtough special relativity does alter the terms you listed, all of which have to do with the energies possessed by particles either by virtue of their motion, or their mass.

For example, in special relativity, electromagnetic interactions can be said to "store potential energy." When two charged particles interact via the Coulomb force for example, there is an interaction energy between them that deserves to be called potential energy just as much as in pre-relativity classical physics. In fact, classical electrodynamics exhibits Lorentz invariance and is in this sense a fully relativistic theory without alteration.

Any other form of energy that is called "potential energy" in a non-relativistic context probably also deserves this designation in a relativistic concept (I, at least, can't think of any counterexamples).

If the source does not move, it creates a static electromagnetic field for a probe charge and it has classical interpretation, kind of $V(vec{r}_1-vec{r}_2)$. If the source moves, its electromagnetic field becomes retarded, so the third Newton law may be violated. In that case there is no potential energy of interaction of particles $V(vec{r}_1-vec{r}_2)$, but there is an interaction of a particle with an "external" electromagnetic field $q_i varphi(vec{r}_i,t)$ that depends on time explicitly.