Is it rude to send a self-correction followup email to professor? I was wondering if I was rude to a professor.

I am a Japanese student at a Japanese university and I'm planning to apply for a British graduate school. Then, I contacted a professor at the graduate school and asked some questions about the school. The professor replied to me very politely. Due to my bad English skills, I misunderstood what the professor told me and wrote something inappropriate. Realizing my mistake 10 minutes after sending my original reply, I sent him a message again, apologizing for my mistake with a correction. I think I did things in a very polite way, but I still worry if I made a bad impression at the same time.
Do you think the professor would see me as a rude student? Please let me know your ideas.

*I mean I made just a simple English mistake. I should've written "my" instead of "your". I did not write anything like insulting him.
Different cultures have different standards of politeness. Educated people tend to understand that. By UK standards you were more polite than you needed to be, but by Japanese standards, perhaps, just right. 

Don't think of it as a mistake. There should be no issue at all. In fact, I'll guess you've made a good impression. 

The opposite problem, in which a person isn't polite enough in the local culture is a bit more of a problem, of course. I hope that if I were to meet you in Japan, you would forgive me for my seeming lack of politeness in some situations imagining (correctly, I hope) that it isn't intended.