My kids are dependents on my visa, I study at Chiang Mai University. I met another family there who also did the same thing so I would suggest emailing them.
Chiang Mai University offers great Thai classes, in person that are really useful and through the program you can obtain a visa. Like everyone said, don't risk doing an online course. The courses are quite fun tbh and they help alot with day to day life in Thailand.
It's called a certificate of Family relations, not a birth certificate. Strange I know but apparently American embassies have provided the certificate in the past for past students.