Writing thank-you email to professor for agreeing to write a LOR? I asked my professor to write a letter of recommendation for me. She said she would in general be happy to do that. I am quite self-conscious and worried, I would like to send an email back and say thanks, but I don't want to sound overly grateful. Since English isn't my first language, I would like to ask you how you would write something like this. Or: Would you maybe not write back at all, since the professor's email box must be super busy and sending a simple thank you email might annoy them more than anything else?

I'm so not sure about this, thanks a lot!
You're over-thinking this. It's almost always appropriate to say thanks when somebody does something for you. You won't be taking up more than a couple of seconds of your professor's time if you send a quick "thanks!"

Suppose that you'd asked for the reference in person rather than by email. You wouldn't say, "This professor is really busy so, the instant they say yes, I'll save two seconds of their time by running immediately from the room without thanking them."