Regarding the Santos-Dumont claims, it is important to note that, like /u/waspocracy pointed out, a lot comes down to definitions.

The Wright Brothers made their first powered, heavier-than-air flights on December 17, 1903. But the didn't proceed to patent anything, and to maintain secrecy, they refused to make any flights in public. In fact, it was the combination of steady and favorable winds, along with the remoteness of Kitty Hawk, NC, that led them to choose Kitty Hawk in the first place.

They had been to Kitty Hawk several years in the past trying out their gliders. It was their experimental flights in gliders at Kitty Hawk that led them to rework the shape and design of the airfoil they were using.

When they finally made their powered flights, they used a catapult to help get the plane to speed and also to account for the fact that they were flying over sand, which obviously doesn't make for the best surface to wheel over with some sort of landing gear.

Word got out that they had made their flights successfully. This was a shock to Europe because at the time Europe considered themselves (especially the French) the leaders in aviation. When it became apparent that there weren't many witnesses, especially aviation-minded witnesses, the Europeans started to doubt them. Still, the Wright Brothers maintained their secrecy, hoping to continually improve their designs until they had a marketable and commercially viable design they could sell; they would then get patents on those designs to protect it; but until they had that marketable design they insisted on secrecy and staying out of the public eye. They continued to run test flights, and even take up the occasional passenger with local witnesses, but secrecy was the name of the game for them.

It wasn't too long after the Wright Brothers had made their flights that the Europeans began their own powered, heavier-than-air flights. Santos-Dumont was among the leaders. He had been a pioneer with balloons lighter-than-air machines, but he then turned his attention to heavier-than-air machines. In a few years he had a working prototype, the 14-Bis. It could get airborne, but it was very awkward to handle and control over it was very problematic. Still, he was airborne in his machine.

It was around this time that the Wright Brothers finally gave in to the requests for flight demonstrations (along with the fact that they felt could finally start marketing their airplane). They went over to Europe and shortly after Santos-Dumont had made his flight and become a hero, the Wright Brothers flew around Paris. Their plane could not only go higher, faster, and remain airborne longer (much, much longer) than Santos-Dumont's 14-Bis, but they could control it.

One of the key issues the Wrights had recognized was that the machine is worthless if you can't control it. They did this through wing warping, and it gave them excellent control. Santos-Dumont had nowhere near that kind of control.

The Europeans were awed by the Wright's machine. The leading aeronautic minds all took back their previous statements of doubt about the Wright's claims of flight in 1903. The Wrights were lauded as the true inventors of heavier-than-air, powered flight. Santos-Dumont was furious. Just weeks earlier, he had been the national hero of France. Now the same people that had been lining up to buy him dinners and treat him were lining up to treat the Wrights.

The one thing that Santos-Dumont did before the Wrights was to take off and land with landing gear wheels. The Wrights had used a catapult and pulley system to get to take off speed, and when they landed the plane basically skidded to a stop.

So some of Santos-Dumont's supporters maintain that this makes him the true pioneer in that he put all the pieces of the puzzle together and that the Wrights were missing that one piece. Personally, I don't buy it, but that's just my opinion.

Source: The Bishop's Boys by Tom D. Crouch

PBS documentary about Santos-Dumont