Disclaimer: The following response is my method of defining these terms, and is not intended to be read as being what I think are the only way to define them, merely one system that I think would work well.
They’re not distinct. I would say that sport is a specific subdivision of game, which is a subdivision of competition, which is a subdivision of activity. Thus all sports are games, and all games are competitions, but the reverse is not true. This is not hard and fast, though, and has some exceptions.
I would define a competition as any activity in which one person’s achievement is judged against another’s. An art exhibit or a county fair is a competion, but not a game or a sport. A job interview is a competition.
A game would be characterized by a standarized set of rules, equipment, and venues, simultaneous or alternating participation by the contestants in the same venue, the ability to directly or indirectly affect your opponent’s play, and a scoring system involving at least some objective component to determine a winner. Monopoly is a game, as is Baseball.
A sport is a subset of game that would involve all of the above conditions, in addition to which the contest would need to primarily involve some test of physical ability, be in strength, speed, stamina, endurance, balance, eye-hand coordination, etc, and must have wide-spread participation.
A couple of things to emphasize in these definitions. To be a game, it would have to involve at least some objective measures, but by no means all.
Given these definitions, let’s examine into which categories some activities would fall. Keep in mind that these are cumulative; by these definitions, all sports are games, and all games are competitions, but not all competitions are games and not all games are sports.
Art exhibit: Competition
Darts: Sport
Chess: Game
Jogging: Physical activity
road racing: sport
cheerleading (at a sporting event): physical activity
cheerleading (at a cheerleading competition): sport
bodybuilding: competition
fishing or hunting(when done competetively): competition
fishing (done for recreation): Physical activity
auto racing: sport
bridge: game
roofball (an activity invented by my brother and I that can only be played at our childhood home): game.
bowling: sport