That the voltage across an inductor is proportional to di/dt (and no longer bothering to mention flux when solving circuits), that's what people usually mean when they say self-induction. The way E.E. is usually taught, one day they tell you that the "induction" that you learned about before is more specifically called "self-induction", to contrast with a new topic that's being introduced, called "mutual induction."
You asked what the magnetic field has to do with the oscillation -- in the LC circuit the energy is continuously swapped back and forth between two different forms, between being stored in the electric field and being stored in the magnetic field. This is analogous to some mechanical devices, like the pendulum, or a mass bouncing on a spring, where mechanical energy gets swapped back and forth between kinetic energy and potential energy. Inertia keeps the mechanical gadget going when the potential is at its minimum. Inductance is compared with inertia.