The terms "slip" and "skid" refer to two different types of uncoordinated turn - neither has much to do with the elevator, instead both depend on what the rudder is doing:
In a skid you have too much rudder input for the turn - the aircraft starts to pivot into the turn.
Because the aircraft is effectively pivoting about its center of lift while flying you can think of a skid as the outside wing advancing (seeing a faster relative wind), while the inside wing is retreating (seeing a slower relative wind). The inside wing thus generates less lift, and the plane starts to bank into the turn.
If you try to correct for that bank using aileron to raise the wing your elevator deflection effectively increases the angle of attack on that low wing, with its reduced relative wind -- do that aggressively enough and the inside (retreating) wing stalls while the outside (advancing) wing is still generating lift, and you enter a spin.
A slip is the same thing in the opposite direction: You have too little rudder input for the turn, and the aircraft is pivoting out of the turn:
In a slip the inside (low) wing is advancing, and the outside wing is retreating, so the inside wing is moving faster, generating more lift, and trying to level the plane. If you try to hold the inside wing down with aileron input you are increasing the angle of attack on the outside (high) wing to keep it up. If the outside wing stalls because of this it will drop the aircraft into something closer to a wings-level attitude, which will also bring you out of the turn.
This is why slips are said to be spin-resistant: The stall encourages the aircraft back into a more stable configuration (coordinated flight) rather than a less-stable one (spin).
The images here are taken from BoldMethod's "Why skids are more dangerous than slips" article, which is a great read and also has a better and more comprehensive explanation of the aerodynamics involved.
My explanation above is a vast oversimplification, but the advancing/retreating wings and difference in relative wind is the one that got the idea to stick in my head so maybe it will help you out too.