Thanks for pointing me pass Jason Cole, Sarcastic. Otherwise I wouldn't have looked at the thread.
Here's my opinion: It doesn't say any CLUBS that designates a player as a Franchise Player for the third time.... Meaning, different ball clubs are thought of as a collective group. It specifically says "any club that designates a player". The club, here the New Orleans Saints, that designates a player as a Franchise Player for the third time has to pay upwards of 144% of the previous contract. In simpler terms, it means if the Saints pick the same player 3 times to be the franchise player. Not, if any team has picked the same player 3 times combined.