Thank you very much for writing to me. No worries, we are engaging in a harmonious discussion and I am really glad to see your point of view.
Firstly, the word, "汉语" is widely used in China not outside of it as mentioned.
Hence as expected you don't see it in our locally published textbooks. i.e. we don't use those words like "汉语" nor "汉人".
We see ourselves as Chinese (华人) not China people (中国人). We speak华语and write 中文. We don't speak "汉语" and are not "汉人".
Hence, when you mentioned "我说汉语", you could have unknowingly offended her because She didn't want to be recognized as a 中国人 when she could be of other nationality. Similarly, because someone is speaking English you assume they are are American when they could be from the UK.
As for the below:
"For your info, in the textbooks for Chinese learners teach us 自行车 but not 脚踏车, 一个苹果 and not 一粒苹果, 我先走 but not 我走先 etc..."
People could accept such differences such but probably not when it comes to nationality matter.
Also the teacher has a part to play in explaining to the learners to be more sensitive when it comes to use of "汉语" and "汉人".
Just my two cents worth.