Is 'restoring force' a particular type of force?

Good question! What you probably haven't been told is that forces can sometimes go by multiple names. There can be a name that describes what produces the force, but also a name that describes how the force is acting, or something else. The term "restoring force" falls in this latter category: it's a name that describes which way the force points, i.e. toward the equilibrium point. The same force might also have another name which describes what produces it. For example, it's quite possible that the tension of a string, or the elastic force of a spring, or gravity, is the restoring force.

In fact, it's possible that multiple forces contribute to the restoring force. "Restoring force" just means the total force that acts toward the equilibrium position. In the case of the bungee cord, the restoring force is the sum of gravity and the elastic force of the cord.

Another case where you might have seen the same thing is with the term "centripetal force," which refers to whatever forces are pointing toward the center of an object's circular motion. Sometimes gravity is the centripetal force, sometimes it's tension, sometimes it's a normal force, etc., or even a combination of different forces.

The restoring force is defined as a force which acts to bring the system back into equilibrium. Total force is just that: total force. For example, if a mass is hung vertically on a spring, then the restoring force is given by Hooke's law: $F_{rest}=kx$, acting upwards. The gravitational force is $F_{grav}=-mg$, acting downwards. The total force is:

$$F_{tot}=F_{rest}+F_{grav}=kx-mg$$