It determines who is the bigger douchebag. If you 1v1 it should be fun, not because 1v1 me fgt ill wreck u. There is no single way of determining if one player is better than another player, it all depends. You can be the most skilled PvPer ever, but if you're a condescending asshole your skill means nothing. A good player, to me, is someone that is good at communicating and using callouts while still playing objectively and holding a decent K/D. But everyone has a different definition of what makes who a better player.

Whomever has better connection is the better player.

No, because being proficient in 1v1 means little in the actual world of online multiplayer. In matchmaking, there are many more factors and variables one must constantly consider. The player that can use these to his advantage and persevere in actual online matches is the better player. A 1v1 match does nothing to reveal which player can do this the best. For example, not every game type is a Deathmatch.

Only when said 1v1 takes place IRL.

It is too easy to kill just once to prove dominance. Whoever can consistently rack up the most kills in PvP I would say is the better player.