Hmm, it seems like a lot of hassle. I guess I can live with it... thanks for the clarification guys.
You do not necessarily need any official documents. What if you are not religious but because of your personal beliefs (not associated with a faith) you object to bearing arms. Anyone can object to bearing arms. It is more advantegeous and easier to be approved to sign the modified oath if you belong to a pacifist faith, however, everyone applying can have a chance to take a modified oath if they so choose. Now, being given a chance and proving are two different cases. In either case, you need to prove your case as to why you would like to take the modified oath.
I assume the same criteria apply as for "conscientious objector" status with Selective Service. This is how they define it:
Beliefs which qualify a registrant for CO status may be religious in nature, but don't have to be. Beliefs may be moral or ethical; however, a man's reasons for not wanting to participate in a war must not be based on politics, expediency, or self-interest. In general, the man's lifestyle prior to making his claim must reflect his current claims.
Modified oath in the court system is usually for someone who is an atheist/agnostic or for any other reason does not want to swear by GOD to tell the truth. One simply swears or promises to tell the truth without referring to God.