Actually there are two reasons for the different spellings. Arabic is written with the Arabic alphabet, and there are various ways to transliterate it into the English alphabet. Different transliterations are spelled differently: assalaam alaikum, assalamu 'alaykam, etc.
The second reason is that this Arabic greeting is also used by Muslims who speak languages other than Arabic and so it's used throughout the middle east, central Asia, parts of Africa, and elsewhere, by people who don't speak Arabic. (As well as by Arabic speakers who speak different dialects of Arabic.) Each has its own different pronunciation, which results in a different spelling in English.
Your spelling of salumu 'aleiko looks like one of these; it might come from a dialect or language that doesn't usually have the letter "m" pronounced at the end of the word. Assalaam alaikum or Salaam alaykum would be difficult for someone to say if their language or dialect never has an "m" sound at the end, so they would change it to Salamu 'aleiko in their pronunciation.