Yes, it's confusing. If Eddieboy's father was not a US citizen, and Eddieboy was born in wedlock, the Eddieboy's mother would have had to have 10 years of physical presence in the US before Eddieboy's birth, including 5 years after turning 14 (assuming Eddieboy was born between 1952 and 1986) for Eddieboy to have US citizenship at birth, and that's hard to meet if Eddieboy's mother never went back to the US after a young age (unless she was the dependent of someone in the US military where time spent as such can be counted as time in the US). So I'm guessing that Eddieboy was born out of wedlock, in which case Eddieboy's mother only had to have been physically present in the US for some continuous period of 1 year, which she could have met as an infant if she left the US after turning 1. (The route to get citizenship via physical presence of a grandparent didn't become available until 1995, and I doubt Eddieboy was under 18 then.)
Anyway, assuming that Eddieboy is a US citizen, I am still not clear on what exactly he (I will assume Eddieboy is male) did at the US embassy and why he had to take an oath there. I can think of 3 possibilities:
1. He is saying that his children were automatically US citizens at birth, and he was getting a CRBA or US passport for them. If this is the case, the children are already US citizens and can apply for US passports at any time. In order for this to be the case, assuming the children's other parent was not a US citizen, Eddieboy would have had to have been physically present in the US for 5 years before the child's birth, including 2 years after Eddieboy turned 14 (since the children were born after 1986).
2. He is saying that he filed N-600K for his children to naturalize under INA 322, using his mother's physical presence in the US, or his own physical presence in the US including both before and after the children's birth. But this is unlikely because 1) the children would need to come to the US to take the oath before turning 18, and 2) either he or his mother would need to have 5 years of physical presence, including 2 years after turning 14.
3. He is saying that he filed I-130 petitions for his children to immigrate to the US (i.e. become green card holders). When they entered the US on their immigrant visas, they became US permanent residents (green card holders), and, if they were under 18 and living in the US with a US citizen parent at the time, they immediately became US citizens under INA 320. If they automatically became US citizens like this, they could simply apply for US passports. They could file N-600 to get Certificates of Citizenship, but this is not necessary. If they were not under 18 when they entered, they did not become citizens under INA 320, and they are still permanent residents today. In that case, they could apply for naturalization on their own if they want with N-400 once they have been permanent residents for 5 years.
