Is it permissible for a female doctor to record online courses to watch them later, knowing that the website prohibits recording, even though she will not share them with anyone out of respect for the publisher's rights?

If the website only prevents registration for not sharing content with others, then there is no harm for you in registering for personal follow-up, as long as you do not share it with anyone. This is confirmed based on the opinion of those who restrict copyrights to commercial use, excluding personal use, or those who deem it permissible absolutely. Nevertheless, it is better to clarify your situation to the website administrators and seek their permission for registration.