at the first time,for the first time,on the first day有什么区别呢?

at the first time 指在第一次
如一个人打算分3次做一件事情,at the first time,he made a ...,but failed.at the second time,...,failed again.
for the first time 指为了第一次,第一次.
如,On her boyfriend's birthday,she cooked for him for the first time.她男朋友生日那天,她第一次为他做饭.
on the first day,指第一天
如一个人打算用3天去做一件事情,on the first day,he made a...,on the second day,he did sth...
明白了吗?