Someone in r/answers wanted actual references for the rules for Princes and Princesses, so I looked them up, and figure people here may be interested as well:

1. In 1917, George V (grandfather of Elizabeth II) issued these Letters Patent (bottom of left column) which established the basic ground rules.

2. In 1948, George VI issued these Letters Patent (second section from bottom left column) which established a special rule for the children of Princess Elizabeth and Prince Philip. This was necessary because the previous rule didn't provide for any title for the children of a princess, and the baby who would become second in line for the throne (Prince Charles) was about to be born.

3. Queen Elizabeth modified the rules with these 2012 Letters Patent. Previously, only the eldest living son (now Prince George) of the eldest son (Prince William) of the Prince of Wales (Prince Charles) would be a Prince. This change ensured that Charlotte would be a Princess.

If you're wondering, the current rules do indeed indicate that if Prince Harry married and had kids while Queen Elizabeth still reigned, his children would not be princes or princesses.