Within Shodokan Aikido, we practice something we call Randori (乱取り) in which two players fight one vs the other. Mostly, it is a tool for learning how to use Aikido versus a strongly resisting opponent. It is a learning tool and not used to determine who's best.
Shiai (試合?) is what we call the next level: it is a determination of who's best in that bout at this time. But there are no referees, no crowd, no medal, and no glory.
Kyousou (競争?) on the other hand is what we call shiai when there are referees, crowds, medals, and glory of winning a tournament.
Are we using all those terms correctly or have we made an utter mess of it?