Could you tell us from where/who you got this information .
From the article below , it seems there is only 1 rule about it which is not dependent on religion , and it is very clear/simple , so I don't think it would have different procedures in these 2 different courts .
My Indonesian wife said that she already heard that divorcing in Pengadilan Negeri/National Court (for non-Muslims) is easier , but I myself divorced (without lawyer) at Pengadilan Agama/Religious Court (for Muslims) and didn't see any difficult .
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From
https://www.hukumonline.com/klinik/...ta-kelahiran-anak-luar-kawin-lt55e08ea842273/
Title : Inclusion of the Father's Name in the Birth Certificate of an Illegitimate Child
By Tri Jata Ayu Pramesti, S.H. , 14 September 2015
... The issue of legal relations between children born out of wedlock and their parents is regulated in Article 43 paragraph (1) of Law no.1 of 1974 (“Marriage Law”).
However, the Constitutional Court (“MK”) through decision Number 46/PUU-VIII/2010 dated February 17, 2012 decided that Article 43 paragraph (1) of the Marriage Law is contrary to the 1945 Constitution if it is not read:
"A child born out of wedlock only has a civil relationship with his/her mother and his/her mother’s family and with a man as his/her father who can be proven based on science and technology and/or other evidence according to the law to have a blood relationship, including a civil relationship with his/her father’s family."
So, if you want the child to have a civil relationship with his/her father too, this can be proven legally, namely through a court ruling. For the purpose of issuing a birth certificate, a court ruling is required for the legal protection of the child himself/herself.
The Republic of Indonesia based on Pancasila and the 1945 Constitution ("UUD 1945") is essentially obliged to provide protection and recognition for the determination of personal status and legal status for every population event and important event experienced by Indonesian citizens. Recognition of legal status in important events here, one of which is the issuance of a birth certificate.[1]
Then, regarding the court that has the authority to issue a ruling on the legitimization of an illegitimate child, for those who are Muslim, the application for a court ruling is submitted to the religious court. The Religious Court has the duty and authority to examine, decide, and resolve cases at the first level between people who are Muslim in the field of marriage.[2] ...
Meanwhile, for those who are non-Muslim, the application for a court ruling on the legitimization of an illegitimate child is submitted to the National Court ...