Indonesia is the largest Muslim country in the world by population, with over 200 million (88.2% of total population) people who identify themselves as Muslim. Indonesian Muslims can be categorized into two main groups: “modernists,” who adhere to orthodox Islam while accepting modern advances and education, and “traditionalists,” who are more apt to follow local religious leaders. (Wikipedia)
History of Islam
Arab Muslim traders brought Islam to Indonesia possibly as early as the 8th century through Banda Aceh. However, Islam did not start to spread until the 13th century. Islam advanced as more of the archipelago increased its trading with the outside world. The religion came into Java from ports in Banten and Batavia. By the end of the 16th century, Islam had spread through all of Sumatra and Java. (Wikipedia)
Muslim Sects
Most Indonesian Muslims are Sunni, although there is a small minority of Shia Muslims. There are also Sufi Muslims in Indonesia. Many Sunni and Shia Muslims do not consider the Sufi to be true Muslims because they use mysticism and believe that they can have a relationship with God. Ahmadiyya make up another Muslim sect in Indonesia. Indonesian Muslim religious leaders do not recognize Ahmadiyya as true Muslims but as a cult; however, there are 242 Ahmadiyya groups in Indonesia.
Folk Islam
Almost all Indonesian Muslims, whether Sunni or Shia, practice what is known as “folk Islam.” Folk Islam mixes animistic practices with traditional Muslim rites. For example, before a house can be built, a chicken must be sacrificed for protection. The chicken’s head, feet, and wings are buried under the foundation stones. The head must face east and be surrounded by rice and spices. The chicken’s blood is spread on the stones and a coin is placed under the cornerstone. Once this is done, the house can be built. Yet, most Indonesians Muslims will also pray five times a day and fast during Ramadan.
There are several Muslim organizations in Indonesia that lead their followers in all areas of Islamic life. The leading “modernist” organization is Muhammadiyah. They have branches all over the country and have schools, orphanages, mosques, and clinics, along with many other public service projects. Nahdlatul Ulama is the largest “traditionalist” Muslim organization. It is also the largest Muslim organization in Indonesia, with 40 million followers. They run the majority of Indonesia’s Islamic boarding schools. There are also several other organizations, such as the Front Pembela Islam, a hard-line Islamic group that is known for violence.
I think you should add the essential presence of Chinese Muslims. They played an important part of spreading Islam in Indonesia. That’s why some of the traditional Islamic custom in Indonesia have similarities with the Islamic custom of the Chinese Hui Muslims (and the tendency of diluting animistic customs with Islam may have come from Hui’s too). This explains why traditional Islamic clothing in Indonesia (baju koko/baju tuikhim) has similarities with Chinese traditional clothing rather than Arabic clothing.