Indonesian have lots of words to call themselves 😆.

There is aku, saya, beta, hamba, gue (even this has many variations of spelling: gua, guwe, gw), ane (from Arabic ana), kami (this is actually plural, but some people use it as singular I don't know why), kita (same like kami), and many others which came from the many local languages.

But between aku and saya, aku is more friendly, while saya is more hmm.... businesslike maybe. You use aku if you are close to the person you are talking to, or if you want them to see you as a friend, or when you respect that person but you want them to like you. You get the idea?

Saya is widely use in office manner, but usually in writing (email, formal letter, etc) because most people still wants to be friends with their colleague, so in conversation they still use aku, well except to a certain boss that they think is very high level and their life (read: salary) depends on him/her.

In conclusion, aku is more laid-back; saya is more serious, distance.

For any formal letter, like letter to government, office, etc, always use saya. If you use aku, the recipient may think that they can treat your letter not very seriously.