I consider a game completed when I finish its main storyline, complete its campaign, or whatever happens to be the case in that particular game. Typically if you beat the final boss and you see the credits roll, then you just beat the game.
As for the main question... I typically beat the games I play eventually. But often, particularly with RPGs from older generations, I would start them and never beat them for many years. Not because they were bad or I got tired of them, but because I would get to the very end, need to go to the final dungeon, and then would instead get bogged down with all the optional side stuff. Getting the golden chocobo in FF7, monster hunting in FFX, and so on. So I get very involved in those things, waste a lot of time, and then get kinda tired of it and put it off for later. Only when later comes, I realize I can't get into a game so near to the end and have to start it over again. Since I don't want to play through the entire thing again so soon, I set it aside for a year or two, and then start it anew, so that the cycle can repeat itself. I have beaten most of those games by now though, and in a few cases even managed the optional portions as well. Oddly this phenomenon mainly only happened with PS1 generation games, so maybe it's just because I was so much younger when I played them.