I think school canes vary quite a lot from school to school and over time. However, 8 mm diameter sounds a little thin to me although boyhood memories are likely to be quite unreliable. I remember our woodwork teacher improvising with a 3-foot length of 3/8 inch dowel (10 mm approx). It was about the right diameter but much too fragile to do the job.
The size (width) and amount of flesh on the buttocks are quite important in determining the effect of the cane. A thin, young teen boy of yesteryear had very little flesh covering his bones when in the classic touch-the-toes position.