1. Today is my birthday. One of my family members knew that. I have a sister. She gave me a watch for my birthday.
(Is this passage grammatical? Does it mean that she didn't give the watch on my birthday. She gave me a watch yesterday for my birthday. Am I right?)
2. She gave me a watch on my birthday.
3. She gave me a watch for my birthday.
(#2 mean she gave me a watch on March 13th, for example. However, In #3 mean she gave me a watch on March 13th or on March 12th or on March 14th. Is that right?)
In #1 and #3, "for my birthday" means that was the reason for the gift. It could have been given ON your birthday or ON another day, but it's not exact.
In #2, "on my birthday" is exact. It was given to you ON the exact day you celebrate your birthday.
1. The passage you provided is grammatical. It states that your sister gave you a watch for your birthday, but it does not specify the exact day when she gave it to you. The passage does not indicate whether she gave it to you on your birthday or on a different day. So it is possible that she gave it to you on your birthday or before or after your birthday.
2. The statement "She gave me a watch on my birthday" implies that your sister presented the watch to you specifically on the day of your birthday. In this case, March 13th. So you are correct in your interpretation.
3. The phrase "She gave me a watch for my birthday" suggests that your sister gave you the watch as a gift in celebration of your birthday. It does not restrict the timing to just your birthday (e.g., March 13th). It could mean she gave it to you on your birthday, or perhaps one day before or after your actual birthday. The specific date is not specified in this statement. Therefore, it is possible that she gave you the watch on March 13th, March 12th, or even March 14th.