An athlete to me is someone that treats/trains from the perspective that they are competing in a sport and enjoy the whole process of training, have a competitors spirit and want to win (no matter how they get the win - a win is a win). They treat MMA as a sport - game plan & strategize and don't necessarily care for being drawn out into wars - they want to win no matter how they go about it. I feel like athletes in MMA probably enjoy the sports aspect of what they do - more than the actual fighting. They're the kind that would end up in any sport - not necessarily fighting. It just happens that in their case the cards that were dealt means they fight for a living.
A fighter though on the other hand is someone that revels in the carnage of a fight like
@j123 said they have a lust for fighting. Sometimes that love of fighting makes them
reckless (inside & out of the cage) because they are drawn into wars of attrition that they might not otherwise have to be drawn into. They don't game-plan as well as athletes do I feel as well - sometimes they abandon their strengths because fighting for them is not about winning - but about how you win. Pride comes into play with this significantly. Also I think that they don't necessarily enjoy the whole process of training but rather enjoy the sparring aspect much more. They're the kind that wouldn't end up in any other sport except a combat sport.
Martial artists to me is someone that trains from the perspective of a lifestyle - they're in it for the long haul not just for the short-term (their professional careers). I feel like a martial artist also doesn't particularly enjoy fighting for living but does so out of the need to make a livelihood. They treat fighting in a similar way to athletes I feel.
I think most are a combination of the above but in differing amounts and probably lean more to one than the other other two. I think there are more athletes in MMA nowadays, followed by fighters than lastly martial artist. Personally I feel like there are very few martial artists - I feel like most genuine martial artists would have an issue beating up other people for entertainment/sports purposes - after all it's more than a sport - it's a way of life - a philosophy.
When I think of an athlete I think of someone like Jon Jones, Tyron Woodley or DJ. When I think of a fighter - I think of Cowboy, Lawler or Hunt. When I think of martial artists - I think of Uriah Hall, Machida or GSP.