I agree that the rules are confusing. Here is an updated link: http://www.renju.net/study/rifrules.php
And another page which may be more helpful http://www.renju.net/study/rules.php
Here is the definition of a three: A row with three stones to which you, without at the same time a five in a row is made, can add one more stone to attain a straight four.
The move 31 N14 is not a double three, it is a four-three which is allowed. A line is a three if it has three and only three stones.
Renju is an enjoyable game but it does take some study and practice to understand the rules.