Would going down a slide be considered sliding friction?

Broadly, there are two 'kinds' of friction: sliding and static.

Static friction is the resistance to motion of an object that is not moving in relation to the surface it is supported by.

Sliding friction is any friction that is encountered by an object already in relative motion.

When I say 'relative' motion, you can take that to mean just 'motion'. But if, say, you were on a train doing 100 mph, and an object were resting on the floor next to you, its relative motion with regard to the train would be zero, so the force of friction would be static friction. If it were moving relative to the floor of the train - lets say it was moving at 95 mph relative to the ground, or backwards at 5mph relative to the floor of the train, then you would consider it to be undergoing sliding friction.

So going (sliding!) down a slide definitely involves sliding friction.

In general, I would define two types of friction static and kinetic. What your book is calling sliding friction I would consider kinetic friction (unless your book has an unnecessarily specific definition of sliding friction). The main difference between the two type of friction is how the objects are moving in relationship to each other. If the objects are not moving in relationship to one another, the friction is static (stationary). If the objects are moving in relationship to one another, the friction is kinetic (sliding).

When sliding down a slide, you are moving relative to the slide. You feel kinetic friction between yourself and the slide. The direction of the force of friction is opposite the direction of motion.

Umm, no, that's not right. The frictional force is static and directed up the ramp. It's not clear what they mean by "rotate in the movement opposite to its translational motion". I could take that to mean the bottom surface of the ball moves opposite the translational motion, i.e. upslope, or that the rotation is opposite of what you'd expect, i.e. the bottom of the ball moves downslope. Anyway, both are wrong. The ball could slip by not rotating at all. A non-rotating ball going down a ramp will be sliding. Or it could slip by not rotating enough. It could also slip by rotating too much, but some outside force or initial condition would have to be involved for that. As the ball rolls down the ramp, it accelerates. In order not to slip, the ball has to experience angular acceleration. Suppose the ramp slopes down to the right. Then the ball needs to accelerate in a clockwise direction. That requires torque. The torque is provided by static frictional force to the left at the bottom of the ball - which is up the slope.

sliding friction is the kinetic friction between moving surfaces

Picture Of Sliding Friction