A swordsman is someone whose swordplay is a primary aspect of their power. Meaning that, while you can be a martial artist and a swordsman, your martial arts skills aren't going to be much higher than your swordplay. If while using swordplay you're losing a fight, and you switch to martial arts because it's your "real power," then you're not a swordsman.
Someone who uses a goddamn sword. And anyone who tries to argue anything else is just an idiot.
Just because Mijawk isn't stronger than Shanks or Fujitora doesn't mean he's not better at using a sword than they are, especially when Mihawk's entire arsenal is reliant upon his swords while Shanks is a Haki monster and Fujitora has his own gravity devil fruit.
That's not what his title means though. It means if you're a swordsman, he's stronger. I still find it hilarious that people think Shanks being a Haki monster means anything; Haki was noted to be very important to swordplay in the New World when the first thing Mihawk did was teach Zoro Haki and told him it was essential because even a chip in his sword is a shame as a swordsman.
People seem to forget that Haki is a supplementary power; it's not a fighting style, it enhances what your fighting style is. So Shanks+Monster Haki still inferior to Mihawk.
Law's fighting style does not revolve around his sword it revolves around his df. Hes uses a sword as a means that is all it is used for. Unless you believe law needs his sword to use his fruit if so then we have nothing else to discuss
This isn't true though. Law's fighting style primarily revolves around both his sword and his DF. Take away one or the other, and Law loses his fighting style. Doflamingo explicitly calls what Law does with his DF as "swordplay." If using a separate power through your sword means you're not a swordsman, then Zoro stopped being a swordsman when he started using Haki with his blades. Law's DF ability is just like Haki on a sword: It's a power that enhances your swordplay. Same with Fujitora.
I thought that too at first but someone pointed out the scan of Fuji eating Ramen while calling down a meteor. As to your first point I'll entertain the possibility Mihawk is stronger than Shanks, it is possible, I just don't believe it will hold true. WSS simply and literally means you can beat anyone while you both are holding a sword but if you're opponent drops that sword and starts fighting via different methods WSS doesn't guarantee a victory in that scenario.
Then you're not a swordsman in the One Piece world because you're just dicking around with a sword. We've seen that in the world of OP, being a swordsman is more than just a fighting style, it's a state of mind almost akin to a culture. No swordsman is so much stronger at another field that switching from one to the other turns the tides entirely.