As others have said, it certainly isn’t rude to ask for a recommendation - as to asking for “a strong recommendation”, it depends on your relationship with the professor. Will t understand that you are not asking for a letter that is stronger than is warranted? If so, to say something like, “For this position, I will probably need a strong recommendation. Would you be able to write such a recommendation for me?” That puts it back on the professor to accept the request or not. (And distinguishes from just asking for a recommendation to just comply with the required three, or something.) But I think all teachers/professors would try to write an honest recommendation - otherwise, it means nothing. If you have been able to show the teacher or professor by your performance that you deserve it to be strong, I doubt whether you asked it that way would change what gets submitted. (And many recommendations these days are online - fill out some form - type of submission anyway, which I never liked. Makes it more difficult to speak to the specific student’s abilities and capabilities. “Out of all the student you have known in your career, this student ranks in the top _______ %” Very awkward, I’ve known a lot of students in forty five years of doing this. This isn’t a good way to identify who this student is.) I have turned down students when asked for a recommendation because I didn’t feel what I knew about them and their performance in my class suggested t were a good candidate for whatever t were applying to.
Many of my applications are not recommended. I get in, but the professor will be less inclined to recommend me if the student does not have a recommendation letter. When I have to tell the professor, he sometimes feels my refusal suggests it will look better if it came from me. If we are talking about the current student, for the past few years, I am not going to recommend a professor when I feel he or she isn’t a good fit for me. So I have to make a determination about whether it would cause any conflict of interest.