​N-600 question

I got my citizenship through naturalization and changed my name from a long Four names structure to just first name and grand father name "Family Name" via immigration court judge.

Was: Alex Bernard Carter Regan
Changed to: Alex Regan

My kids under 18 years are automatically US citizens. I am in the process of applying for N-600 Certificate of Citizenship for both of my kids. However, I need to change their last name on the application from Four names structure long to just first name and same as my grand father name "Family Name".

Was: Hannah Alex Bernard Carter
Would like to change to: Hannah Regan

I was told you can change the name structure on form N-600 on line one "Current Legal Name" and attach copy of my court legal name change letter that came with my US citizenship certificate.

Is this correct procedure to do so?

Please advise.