Basically, transfer system is based in 2 different filters:
First, the game choses a "X" value that is used to determinate how higher must be a transfer fee to filter out other manager's offers amd then it compares wage offers to decide who takes the player. In your situation, your "X" value was 10%, but the system decided that the "X" value needed to be higher than 10, so you had to compete with other manager's offers, and the player picked the higher wage offer.
In other words, if you want to grab a player, you must offer the higher transfer fee and the higher wage as possible, sometimes you don't need to compete in both: If you want to avoid a wage war, you must offer far more than an extra 10% extra as transfer fee.
