The key to the conundrum is that for the purpose of explaining the apparent forces on someone to whom a rotating frame of reference appears to define stationary, for example all human beings everywhere, centrifugal force may need to be taken into consideration since it appears to be there. Although it may be small depending on the speed of rotation. Which is what your girlfriend's textbook says. For the purpose of stating Newton's laws of motion in an inertial frame of reference, which is what your teachers were doing, there is no such thing as this centrifugal force. You might reasonably think that since they contradict, one of those points of view must be so stupid that nobody would ever say it. But that's not the case.
"Centrifugal force" and "Coriolis force" exist as terms in the equation of motion of an object relative to a rotating frame of reference:
- Consider "rotating with the frame of reference" to be stationary. That's just what "frame of reference" means.
- Consider an object initially "at rest" (that is to say, at some instant in time it is rotating with the frame of reference), but nothing is in place to keep it rotating. Like spinning something on a string in a circle and then releasing just at the instant we start to calculate its motion.
- Let time run from that initial point.
Then the object moves away from the centre of rotation. In an inertial (non-rotating) frame of reference it was not initially at rest. In that frame of reference we'd say that it moves in a straight line. In the rotating frame of reference it accelerates away from the centre -- the initial acceleration is directly away but it starts to follow a curved path.
The "force" that causes the initial "acceleration" is called "centrifugal force", and the "force" that acts on a moving body in a rotating frame and causes the the curve is called "coriolis force".
But the equations of motion in a rotating frame of reference are horrible, and the equations of motion in an inertial frame of reference are really simple. So, who cares about rotating frames of reference, to the point of giving a name to a "force" that doesn't exist in the inertial frames of reference that we prefer for calculation? People who live on a planet, is who. These non-existent forces have to be taken into account if you want to accurately compute the flight of a sufficiently long-range artillery shell or the movement of weather, relative to the ground instead of relative to some fixed point in space through which the earth rotates.
Do they exist? If you take a rotating frame of reference then they can be observed, just like any other force, and for that matter we subjectively experience them when we spin fast enough. If you take an inertial frame of reference then there's no such term in the laws of physics. So yes they exist, you can measure them if you're standing on a planet. No, they don't really exist, they're just a by-product of the frame of reference you chose. A bit like gravity in general relativity is a by-product of choosing an "unnatural" frame of reference, one that fails to follow the curvature of space-time ;-)
Note that in both frames of reference, inertial or rotating, an object that remains stationary in the rotating frame (and rotates in a circle in the inertial frame) necessarily experiences a "centripetal force" (a force towards the centre). Thus gravity causes things to orbit, and the tension in a piece of string causes a spinning poi to follow a circular path. So what "centrifugal force" really is, in a rotating frame of reference, is the term you need in order to satisfy the requirement that a stationary object must experience 0 net force. In an inertial frame that object describing a circle is not stationary, so it does not experience 0 net force, so there is no balancing term.
You are on to something when you speak about equal and opposite forces. Since the spinning object experiences a centripetal force, the object exerting that force necessarily must experience an equal and opposite force. The moon pulls the earth in its direction, and the string of the poi pulls your hand in the direction of the poi. This is not what is usually called "centrifugal force", but it is away from the centre, and it "really" does exist.