Should you use the person's original name or their new name on the record?
You should use an Or between the original name and the alias. On all of the examples in the project instructions, they do show to use the OR - except the first example which is How to Index a Petition. It is pretty obvious that James Andrews is also Dimitrios Andriakos. It should have been indexed as
Given Name: Dimitrios or James
Surname: Andriakos or Andrews
The DEC and PET combined record is off too. Ignashak is the surname: Ignashak or Anders. I think the reason they put Ignashak in the given name field is because it goes along with the note: Note: If you cannot determine if a name is a given name or a surname, type it in the Given Names field. The same applies for the example How to Index a Naturalization Name Change with Marie Rosa Flores De Torres. Her maiden name was obviously Flores...so why index the name Flores in the Given Names field and AGAIN as a Surname Flores or DeTorres. Bad examples.
For DEC/PET, they are taking from the PET because the PET is the second step to becoming a citizen, therefore Anders is the correct surname.
You should use the Or when variations of names that person used is on the card. They were both "correct surnames" at some point in their life.