There are two different but connected motions one can speak of when discussing pulse and wave propagation. I suspect you may not have a clear separation of these two ideas.
One type of motion is the disturbance motion or wave motion. When you watch a wave move, this is typically what catches your eye. If you see a wave and you watch it travel, you're watching the disturbance travel. There's not actually a physical object that travels, but it's the disturbance itself that does.
Another type of motion is the particle motion. This is the motion that each individual bit of material is undergoing. Imagine you tie a small bit of string to a slinky. The particle motion is what the bit of string does.
The key idea is that for a longitudinal wave, the particle motion will be "forward and backward" while the motion of the wave will only be "forward".
Take a look at this video of a longitudinal wave. Try to identify the two different types of motion. You'll see the backward and forward motion of the "particles", as well as the forward-only motion of the disturbances.
If you're still unsure about this, that video link ends with a transverse wave. In those waves, the particle motion is actually perpendicular to the wave motion. If the wave is moving to the right, the particles can be moving up and down. This might actually be an easier example for separating the two types of motion.