I myself like to go through it from start to finish without dying once, usually on the hardest difficulty. Using Doom II as an example, I'd consider it done when I've gone through all 30 non-hidden levels without dying a single time on UV (this is easy for me to do, I'm just using it as an example). Saving is allowed when I'm done for a session. If I die I'd start all over again from the beginning.
I love playing games like this because I feel it results in true mastery of the game and results in lots of replay value.
>I myself like to go through it from start to finish without dying once
I only do this with games I really liked playing through. Basically the same reason as you, because there's a certain satisfaction of mastering a game (even though I'd consider speed-running the true way of mastering a game).
But as for myself I haven't done this with a lot of games. I've done it with Contra Hard Corps,
Contra Shattered Soldier, Shock Troopers, Metal Slug 1 & X . And I've done it more than one time to assure myself it wasn't because of "luck".
I have a pretty completionist play style, so I usually find almost everything in a game as I'm playing it anyway.
I'll consider most games "finished" when I've beat the main gameplay. If I miss a few things here or there, eh whatever.
However, a certain few games I will go autistic and unlock literally everything, like SotN where I've seen all the endings and gotten full 200.6% map completion, bought and used every weapon and item, killed every enemy many times. There is literally nothing in the game I haven't seen or done.
I feel like I've "completed" a game once I've done everything to meet some standard for 100% completion, like collecting all missable items, completing all sidequests, etc. If it's an RPG, I may even go as far as min-maxing character stats.
It's impossible for me to put a game down forever though. Even if I've satisfied conditions for 100% completion, I'll start up a new save file and try to beat the game with self-imposed challenges, like never leveling up my characters, or restricting characters to initial equipment. I've done this with a number of Zelda games already. It really puts the game into a whole new perspective because you have to use entirely different strategies to survive.