If it is said that Moshe Rabbeinu was put into the basket on 6th Sivan after 3 months of his birth, won't his birthday be 6th Adar?
This is a simple question of a principle that is called עד ועד בכלל or עד ולא עד בכלל.
If you start counting from the 7th of Adar, then three months out is reached on the 6th of Sivan.
If you start counting from the first completed day, meaning it begins from the 8th of Adar, then the 3rd complete month is reached on the 7th of Sivan.
If there are too many days for you to keep track of and to understand what is happening, try looking at a chart for counting the Omer where we count the days and weeks.
If you start counting on Yom Rishon (Sunday), you reach the first full cycle of 7 days and 1 week on Shabbat, not the 2nd Yom Rishon.