Hello Aleksandr Both are correct- "Brush my teeth" and "Clean my teeth". You can continue using "brush" as it is commonly used and more descriptive.

It's not incorrect to say "clean your teeth" but I've never heard this from native English speakers in normal conversation in America. At home we say "I'm going to brush my teeth." We tell our kids, "Go brush your teeth!" If they just ate chocolate we might add, "make sure you get them really clean!" However, when your dentist calls to schedule a checkup, you will hear "it's time to schedule a cleaning." It's very normal to say, "I just had my teeth cleaned at the dentist" ("dentist's office" is understood) or even "I just had my teeth cleaned" and it is understood that it was at a dentist's office.

In North American English, a person brushes their teeth at home with a toothbrush, and a dental technician cleans a patient's teeth at a clinic using specialized tools. What is the name of the textbook? Where is it from?

They're both correct. I believe brush is more prevalent in North America than clean. I often hear clean in British and Australian English.

Both are correct, different phrases are common in different parts of the world. English is a global language :)