Both are true. You are born on the first day the wheel turns to Scorpio, but because your birthday is on the day of the change, you is referred to as someone who is on the "cusp." This means you will tend to have traits from both the Libra and Scorpio families. Every individual is different. My birthday is November 20th, which is close to the cusp of Scorpio and Sagittarius, but while I am pretty optimistic for a Scorpio, I tend to personally have VERY Scorpio-type traits. The more research you do about your two houses, the more you will be able to identify what parts of your personality identify with each.
is a good place to get the basics.