It is almost certainly not a cultural thing in the sense that it is encouraged. I come from a university in Asia, and plagiarism here is extremely strict - quite possibly stricter than in the West. Students fail FIRST TIME. There are no second chances. Plagiarism as far as I know is a well known fact to be a hindrance to academia. It may be possible that they think they can get away with it because it is not strict in your institution. That being said, I do not come from the mainland China, and can only speak for the university I am in.

So why does it seem like only Chinese students or Asian ones have the reputation? What you suggest does seem to play a part, but there may be other reasons. I would say it is cultural, but slightly different from what you are suggesting. I think that it is because they are pressured to succeed, and at the same time their inherent abilities (English, critical thinking skills) are not good enough to compete at the highest level, which is why they plagiarize. I am sure that they DO understand why it is bad. These students are either extremely wealthy or extremely poor and managed to get in based on academic merit. The pressures they feel are USUALLY immense, and this may be perhaps why they plagiarise. Of course, there are those that plagiarize based on laziness or whatnot, but you must bear in mind that there may be those who plagiarize for (wrongly) other reasons. My university also has many Chinese students. However, as the regulations are enforced so vigorously, there are almost no instances of plagiarism that I have heard of. So I would say give them one chance, be VERY clear, and if they do it again then they must suffer the consequences. Babying would hurt them.

I think your current one or two chances policy should be enforced. Tell them that it is wrong, and do not let cultural issues get in the way. I believe in the workplace nobody would discriminate based on race or culture, and they must be taught that breaking the rules is not a good thing. I wouldn't be too harsh based on cultural differences, but you cannot be totally lenient as this will ruin them as well as enforce negative stereotypes, which would hurt those from the culture who play by the rules and excel correctly.

EDITS: words and changed content slightly