As sabre23t says, Malaysians (and Singaporeans) refer to themselves by our race since each country has a diverse racial population. I think that the only time we think of ourselves as South-east Asian is if it's related to ASEAN or you're from an international school. There are many countries in South-east Asia and it can get confusing exactly what race you are too. Everyone's great great grandfather or grandmother seems to have once been a migrant from Asia or Europe. This is due to European colonialism in South-east Asia from years past.

South-east Asia also houses a significant amount of Chinese people (though they are a minority in everywhere except Singapore) who can be divided into different groups. There are the migrants from China who came down pre-colonialism as they escaped prosecution there. There's also a story about a Tang dynasty princess being sent to Malaysia as a gift or something. Anyway, these people fitted themselves in with the indigenious people (there are all kinds of South-east Asians too but I won't elaborate here). They are commonly known as Peranakans and have had a long long stay in South-east Asia that they are not really like the Chinese in China (or, for that matter, anywhere else). They've taken on many elements in their adopted country and have gone more 'native' of the Chinese here (but they are natives here so I suppose it's a redundant point?).

Another group of South-east Asian Chinese are migrants who came during the period of European colonialism starting from the 19th century or so. Most of the were male workers, working as coolies and rickshaw pullers. They probably never planned to stay in South-east Asia but stayed anyway in the end for various reasons. I think that this is why the Chinese clans in South-east Asia remain strong as the Chinese are very strongly tied to their dialect groups. Of course, not all of them remained blue-collar workers, and, in fact, many became major businessmen.

Finally, there are the modern day Chinese migrants from China, Taiwan and Hong Kong. Most come for work reasons and decided to stay. I know my parents did.

Note: There are probably Chinese who came through the Chinese-Vietnamese border though I don't know their history. But they're there. Anyway...

When you refer to South-east Asian Chinese, be aware there are many kinds of Chinese. And when you refer to South-east Asians, it also encompasses other races other than Chinese as Indians, Malays, Arabs, native Thai/Vietnamese/Fillipino/etc, Indonesians, etc not to mention that every other person's bloodline is probably 'mixed' and you get Peranakans and Eurasians. We're a mixed bunch here.

So! Asians are most definitely NOT all Chinese. Saying Asians meaning Chinese is like saying monotheists are all Christians. Or Jewish. Or Muslim. Or whatever.