FFF, that is exactly my point; we make the subject complicated when it is not. It matters not whether the slip is "forward" or "side". All we need to teach is how to slip, and what to look out for. For example, when I set up a slip I note the IAS at the point the slip commences and maintain that, it might be lower or higher than the original speed prior to the slip, depending on which side the pitot is located, but by maintaining the speed, you know you are safe and don't need to work out what it should be.
Put in bank then use rudder to maintain the desired direction of flight, whether that is toward the runway or in a turn. See the sink, what a thrill. Easy to get into, easy to fly, and easy to recover, provided you don't get all lost in theory. If you want the theory, do it on the ground.
On the runway, who cares what the maneouvre is called, you don't need to know it is a slip. Just a mechanical process of keeping the fuselage aligned with the runway heading and enough bank to stay on the centreline. Watch out for too much bank, don't want to scrape a wing or pod, and if the crosswind is too great (can't maintain centreline), go around. It is an unnecessary complication to describe it as a slip, it is a wing down crosswind landing is what it is. Sure you are slipping but who cares. That is not the point of the exercise.
So there is only one slip, and even a skid is really a slip, since by definition the controls are "crossed" which is all you need to identify it. If you are not balanced, you are slipping.
Getting lost here, feel free to step in!
What was my point? There is no need to identify slips by when they are being used. A slip is a slip end of story. We confuse students by making them learn stuff they don't need to know, and frighten them to boot. Better to keep it as a practical exercise in how to fly, and keep the crosswind landing as a landing exercise, not a slip exercise.
Now I need someone who is an expert (or at least has been published) to agree or disagree with me, hello out there?