Having my name called out in assembly to report to the Headmaster afterwards only happened to me once and was for me, I think the worst of all. As someone wrote earlier you feel everyone’s eyes boring into you. This was very rare though, we were usually sent by a teacher or prefect, summoned via a note in the class register, or, sometimes one of the school secretaries was sent to fetch us.
The Headmaster, two senior masters, and the deputy headmistress (for girls) were all entitled to use the cane. The canes and punishment book were kept in a tall cupboard at the back of the secretarial area outside the headmaster’s office. Presumably so they were accessible to all of the above staff. The senior masters shared an office on the far side of the school, the deputy and the head were in the main block at opposite ends of a long corridor.