According to the naming convention page WP:NCGAMESDAB, "For video game players, use either '(gamer)' or '(video game player)'". I made a discussion on the talk page there and this still appears to be consensus (albeit a local one), with (video gamer) as a third option based on precedence, the above discussion, and the unclosed RfC at Wikipedia talk:WikiProject Video games/Archive 117#Naming conventions for esports/ pro video gaming people after which Inzo created rule #11 specifically proscribing only "gamer" and "video game player". All but one of these above already have (video gamer) parenthetical redirects due to moves by User:Prisencolin citing WP:RECOGNIZABLE, WP:COMMONNAME, and WP:ASTONISH, although the applicability of those to parentheticals even above WP:NATURAL (i.e. using their real names) is dubious. However, they were moved before the introduction of the guideline by Inzo as "a fairly conservative change on the point" and were promoted by Prisencolin, among others, in the archived disambiguation discussion above.

Currently, the options are:

1. Move (League of Legends player) articles to (gamer) – supported by guidelines, but there are exactly 3 articles with this parenthetical (see breakdown below).
2. Move (League of Legends player) articles to (video game player) – supported by guidelines, but there are only 6 articles with this parenthetical (see breakdown).
3. Move (League of Legends player) articles to (video gamer) – technically not supported by the guidelines, but in the words of the person who added the guideline himself, User:Izno, "I doubt anyone would be bothered by it." This was presumably the consensus both before the technically unclosed RfC and before Prisencolin's mass move.
4. Move Faker (video game player) to Faker (League of Legends player) – this is supported by Wikipedia:Naming conventions (sportspeople), which covers mental games as well, although this does not seem to have been cited in previous arguments.
5. Move articles to their real names per if their pseudonym isn't unambiguous and their real name is unambiguous – this is supported by WP:NATURAL, i.e. Armada (video gamer) → Adam Lindgren. This isn't even mentioned as a choice at the RfC#Disambiguation presumably because WP:NATURAL should be applied if they have no competitors for being the primary topic of their real name given it is not "obscure", although it seems User:DarthBotto was the only one to make even a brief reference to it. Nevertheless this was opposed by Prisencolin above and in the RfC and SnowFire above per WP:RECOGNIZABLE, insisting on keeping the parenthetical instead of a natural disambiguation due to Faker and others being, to quote SnowFire, "only notable under those names", and "'more encyclopedic' is not an argument; usage is an argument".
6. Move to <first name> "<nickname>" <last name> – as in Park "Lyn" June. This was option 4 in the RfC#Naming and was supported by Prisencolin. User:Satellizer and User:SMcCandlish noted that this violates both WP:STAGENAME and WP:NICKNAME, yet the article persists. This may also satisfy recognizability concerns if we want to ignore those two rules above based on consensus.
7. Unbundle and discuss the articles "vis-à-vis their sources individually" – per User:Czar in his move close at Talk:Park "Lyn" June#Requested move 25 March 2016. This seems to mean a case-by-case judging of WP:RECOGNIZABLE which Blueboar also confirmed. However, this is largely in response to most of the listed articles having been moved undiscussed by Prisencolin from (video gamer) to (specific game player) when there is no ambiguity (as there is in Karma (Call of Duty player) and Karma (Halo player), rather than to or from full names or pseudonyms which may very well require separate discussions.

I have limited this to League of Legends players, but this could reasonably be expanded to all articles in the following list of all the eSports/video game players with parenthetical disambiguations.