I personally think there's a difference between being competitive and not liking to lose. Well, no one likes to lose, but you know what I mean.
When you're competitive, you aim to be the best. You're going to lose sometimes, and it's how you handle that, that determines if you're competitive, in my opinion. A competitive person will take each loss, and try to improve themselves through it, while someone who isn't will get angry and film themselves punching a hole through the wall for youtube. Everyone gets bummed out when they lose, but when winning is the only option and defeat leaves you raging hardcore, I just think that's being a sore loser. Being competitive is all about taking your loses, learning from them, and hopping right back in there to try to learn some more. Well, at least in my opinion.
Taking what I said above, I'm an extremely competitive person when I'm into something. Back when I was really into competitive battling, I would save just about all my replays, whether I won or not, and study them to see what I could have done differently. If I lost, I determined the move that lost me the match, and made a mental note to not do similar stuff in the future. I learned from my mistakes, and hopped right back in. Same thing goes whenever I lose/die in LoL. If I'm playing ADC and die in a team fight, I'll reflect on what I did, and how the enemies were able to latch onto me. I could complain that my team didn't peel for me, or whatever, but it's my fault initially for putting myself in a position where I could be focused, so I'll avoid making those mistakes in the future.
Now put me in a game where I don't care, and I won't really try. I'll try to win, yes, but I won't reflect on mistakes or try to be the best I can be. Instead, I'll just have fun.