"Also, I didn't think a TM30 was needed for a 30-day extension, but she gave me what looks like a TM30 form, "Notification.... Residence". "
Yes, it is needed, as I have stated multiple times before. You need to do it within 72 hours of arrival or you get a nice 1,600 baht fine from Jomtien Immigration. Others here have said it's not needed, but as you can see, it is definitely needed at Jomtien.