I think it's fair to say someone invented a thing if they design and build the first one that actually works. The concept of an airplane had been around for a while by 1903, but nobody had one. Similarly today we have the concept of a warp drive. We even have that concept outside the realm of science fiction now, but we're far from having a working prototype and don't even know if a practical example can be built.
It's also fair to debate what qualifies as an airplane. If I were to list criteria, I would include two that the first Wright Flyer did not demonstrate: the ability to take off from level ground under its own power, and sufficient maneuverability to return to and land at the point of departure.
I had the impression it took off downhill for that flight, failing my criterion, but it actually took off from level ground with a strong headwind. I'll give it partial credit; I don't think its rail would have been long enough in calm wind.
The Flyer was on a rail with a wheeled cart. The fact that it moved forward under its own power (faster than Orville could run) demonstrated that it could take off in a calm, just that the rail would have had to be longer.
Their experiments with it ended when a gust flipped it on the ground and broke it beyond repair.
It's also fair to debate what qualifies as an airplane. If I were to list criteria, I would include two that the first Wright Flyer did not demonstrate: the ability to take off from level ground under its own power, and sufficient maneuverability to return to and land at the point of departure.