The concept of Bangsa (People) has been intermixed with Kaum (race) in casual speech that it muddles our understanding of what it actually is.
In Malaysia, Bangsa denotes the collective people who are citizens of Malaysia. Therefore, Bangsa Malaysia refers to all of us Malaysians. This was made officially in 1990 by Mahathir, to securely place everyone under one collective term. It's a Civic Nation concept.
Under Bangsa Malaysia, we have race or ethnic groups, in BM called KAUM. Hence we have kaum Melayu, Cina, India, Kadazan etc.
So where do Bangsa Melayu come from? Well, this goes waaaaay decades before the formation of Malaysia. We used to have Tanah Melayu or Malaya, hence the people of Tanah Melayu (who are predominantly Malays) are called Bangsa Melayu. This is an Ethno-Cultural Nation concept.
Having said that, there are many terms in our constitution that was not updated or streamlined, which made it even more confusing.
Except that Melayu is not the dominant race of Tanah Melayu.
Even after independence Melayu is only like 40% of the population if I remembered correctly.