What is the difference between centripetal and centrifugal forces and list examples of each?

A student in my office temporarily confused the words centripetal and centrifugal. This started me thinking about these two words. They mean different things, but do sound and look similar. I have previously talked about the difference between fake and non-fake forces, but let me quickly define these two:

• Centripetal: This is the force needed to make something move in a circle. The force could actually be a number of things such as: friction, gravity, tension in a rope or any combination. Centripetal force is a name for a real force that has the role of making something move in a circle. This force is always directed towards the center of the circle of motion.
• Centrifugal: This is a fictitious force needed to make a non-inertial (accelerating) reference frame seem like it is not accelerating. This fake force is what it "seems" like pushes you away from the center of the circle of motion.

So, the two key differences: centripetal is real and pushing towards the center of the circle. Centrifugal is fake and pushes away from the center. To look up their word-origins, I used the Online Etymology Dictionary. First I looked up "centripetal".

Centripetal: coined 1687 by Isaac Newton from the latin centri meaning "center" and petere "to fall, rush out" (see petition)

And here is the origin for "centrifugal".

Centrifugal: coined 1687 by Isaac Newton from the latin centri and fugere meaning "to flee" (like fugitive)

The problem is that these two origins seem to say the same thing. Center and fall or flee, which I interpret as "away from". Not good. If you look at the other interpretations of petition, it says "a request, solicitation". Also, "to require, seek, go forward". This is better. Centripetal force is the force REQUIRED for circular motion. Centrifugal force is the force that makes something flee from the center.