I ‘bow’ upon the entrance and exit to my dojo, to my sinsei, and to my sparring opponent (partner) out of ‘respect’....as just a small ‘courtsey’ from the waist. Nothing more...nothing less.
and always with eyes up and head up - never looking down. Otherwise sensei would take advantage of that posture to teach you a lesson.
The key difference is that you're not the President of the United States, and your sensei is not the God-Emperor of Japan.
As it should be, but you aren’t the President of the United States. You aren’t representing 300 million free people.