I once read somewhere that jet fighter pilots used to say something along the lines of: "you're not a true pilot until you've landed on an aircraft carrier, at night, during a storm".
Using this analogy, one might use the criterion of "a really hard task" as the indicator of being a "true X". Of course now we arrive at a notion of "what is a really hard task?". To each their own, but let's use the only fairly objective piece of data I have access to: Steam achievements. Assuming we're talking only about "classic" games, let's forget about Doom 3 for the sake of this argument. Out of all the people who bought "Doom 3: BFG", only 0.9% managed to get "DOOM II: Superior Firepower" achievement, which requires you to complete all levels of Doom II on Ultra-violence. Therefore, if you have ever completed all levels of Doom II on Ultra-violence, you are in the 99th percentile of all the people who bought "Doom 3: BFG" on steam. Being in the 99th percentile usually means you did something "really hard" compared to others, so yeah, there is your "really hard task"....
... but everything I've written thus far is bullshit and I am aware of it. The reason pilots invented such a criterion was because it was a hard task, but ALSO because it was required of you as a pilot. When the airforce says "we need pilots" it's obvious they mean "we need people who can land on a carrier, at night, during a storm". It's a possible situation, it may happen and you are EXPECTED to handle it as a part of your job.
No such thing applies to Doom. It's just a computer game, a hobby. Nobody suffers if you don't know how to play it. Nobody EXPECTS of you to have any sort of a pre-set skill level.
However, there is one group of people that expects something of you in terms of Doom: the developer team. I think it's a safe assumption that if you play the game and you enjoy it, you are expected to buy it. Consider such a situation: if you asked a member of developer team / publisher whom he considered "a true doomer" and gave him two alternatives: a guy who set a world record for a full game nightmare speedrun on his pirated copy, OR a guy who still can't get past the first Revenant, but enjoys the game and bought it, who in your opinion would the said developer choose? I'm inclined towards the second option, I know I would choose it in their place.
I think it's actually something Metallica has said once: that they don't care about fans who listen to their music but do so from pirated CDs. They appreciate the fans who listen to their music AND bought original records.
I'm not saying people who have pirate copies are not "true doomers", but I think that's as objective a criterion as you can get: a person who enjoyed the game so much that they decided to spend money on it when they had alternatives to choose from.