you may be confusing perspective distortion with barrel distortion.
Perspective distortion is not truly caused by the lens, it is determined by the distance between the camera and the subject. Even with your own eyes, you can see this distortion if you get up close. Look at yourself in the mirror using only one eye, then put your face right up close to the mirror. You will see perspective distort your face as you get closer.
Barrel distortion/pincushion distortion are absolutely caused by the lens, you can avoid them by using good primes or by keeping your zooms in their "sweet spot," usually somewhere in the middle of their zoom range.