Malaysian Chinese or Chinese Malaysian?

Following Howsy's post "Oh...you're from Malaysia...so you're a Malay then?" and through the comments, we somehow reach the topic of the definition of Chinese in Malaysia:

Are we Malaysian Chinese or Chinese Malaysian?

This question has been in my head since I left Malaysia when I was 13 years old. Looking at overseas Chinese all around the world, there seem to be a tendency for most people to say their nationality first before their ethnicity. For example:

• American-born Chinese (ABC)
• British-born Chinese (BBC)
• Singaporean Chinese
• Indonesian Chinese
• Filipino Chinese
• Overseas Chinese, etc

Why must nationality come before ethnicity when questioned? Most people do so because that's the way they are taught since young. And some want to emphasise their nationality before their ethnicity when around other ethnic Chinese of different nationalities. But how many of us actually say it because we are proud of our country and so would emphasise the word Malaysian before our ethnicity? What about the usage of Chinese Malaysian? With China currently being a major economic force in the world and everyone wanting a piece of the action, there's a trend going round nowadays with naming your ethnicity before nationality e.g. Chinese American, Chinese Canadian, etc. Some people do it as well because they are proud to be Chinese but are we the "overseas" Chinese recognised by mainland Chinese as "Chinese"? Please leave your comments. Thanks!