Mathematically speaking, "interactions" in SM correspond to Lagrangian terms which include more than one field.
https://en.wikipedia.org/wiki/Standard_Model_(mathematical_formulation)
(the terms which include only one field are "kinetic terms", they describe propagation of particles through space without interaction, such as electron moving through empty space. Essentially, "electron interacts with itself" in such a way that it simply moves in a straight line with constant speed.
There are two "one field" terms which describe self-interaction of Higgs field which can't be seen as propagation; instead, they are responsible for phi=0 not being a minimum of energy).
Let's take the inventory of interaction terms:
- B*psi term (weak hypercharge field * fermion field)
- W*psi term (weak isospin field * fermion field)
- G*q term (gluon field * quark field)
- B*phi term (weak hypercharge field * Higgs field)
- W*phi term (weak isospin field * Higgs field)
- phi*psi term (Higgs field * fermion field)
Gravity does not exist in SM.
Thus we have six interactions in SM.