A more simple example might serve to make this clear:
consider a train that is accelerating with acceleration a. You are on this train and you drop a ball. The only force due to a physical interaction acting on the ball is that of gravity which is due to the ball's interaction with the Earth.
However, you see the ball accelerating away from you with the same magnitude of acceleration as that of the acceleration of the train. Thus if we consider the acceleration of your reference frame (keep in mind that since you're measuring the motion of the ball with your position as the origin, you are at rest in your reference frame) as the source of a force on the ball, it applies a force of magnitude ma, where m is the mass of the ball. This force isn't due to any physical interaction that you can observe, that's why we call it fictitious. I agree that calling it fictitious is confusing and that inertial is less confusing and also an apt name in that the magnitude of the force is proportional to the mass of the object
A similar thing happens when you're in a rotating reference frame (for instance say someone does this while on a merry go round) except that now this fictitious force is directed outwards from the center of the merry go round. This type of fictious force due to being on an accelerated (or noninertial reference frame) is called centrifugal.
A centripetal force is the net force in the radial direction due to real forces and is generally talked about in an inertial or non-accelerated reference frame. Hope that clears it up a bit