I found this definition of a splitting field but I am wondering if the second condition does not implies the first one.
If $L$ is generated over $K$ by the zeros of the polynomials of the family, does not it follows that such zeros belong to $L$ and so every $f_i$ splits into linear factors?
If (2) does not hold, there would be infinitely many "splitting fields". The condition (2) guarantees that L is the smallest field in which the function family factors into linear factors.