Malayan is built on the place name Malaya, so the word carries that older, historical flavour: our sources tie it to the former Malaya or to Peninsular Malaysia, whatever family a person comes from. You meet it most in writing about the past — the rubber farms, the railway, the years before the country took its present shape. For a person who holds Malaysian nationality today, Malaysian is the usual word. It is a proper noun, so the capital M never drops, wherever it sits in the sentence.

Here Malayan is the name of a language, not a person. Language names in English take no a and no plural, so people speak Malayan, read Malayan, and learn Malayan — never speak a Malayan. The capital M stays, as it does with English, Japanese and every other language name. The language is today more usually called Malay, and that is the name you will meet more often.