How about we come to an agreement that "Pokémon Trainer" is Red and he isn't at the same time?
Going literally by FR/LG, the player character has no "real" name. Unlike cases like Link, Ness, etc., there was no defined name for the Player from Kanto. It's more or less like Villager in that regard.
"Red" is just one of many names you can choose from in (obviously) Pokémon Red, Pokémon FireRed, and Pokémon LeafGreen. In Pokémon Blue, Pokémon Green, and Pokémon Yellow, this option is replaced by the respective colors.
Hell, this option is present for both the male and female Players in FR/LG.
However, it is still a fact that the character has been established as "Red" in NPC appearances; G/S/C, HG/SS, and B2/W2. It also is worth noting that within dummy data for FR/LG, the male player is known as "Red" while the female player is known as "Leaf" (in fact, this is where the common name for her in the fandom comes from). In other media such as various manga and Origins, the character is also given the "Red" moniker, further establishing the correlation.
Though it isn't the first "name" he was given.
Prior to G/S/C, the Kanto player was referred to in outside media by GameFreak as "Satoshi", the 2nd name you could choose from in Pokémon Red after "Red", named after Satoshi Tajiri and is the native Japanese name for Ash Ketchum. Sakurai even uses this name to refer to him on his pre-Melee poll.
In the original Pocket Monster Red and Pocket Monster Green, the unused default name for him was "Yamaguchi". In Pocket Monster Blue, it was "Geifuri", an abbreviation of "Game Freak", and in Pocket Monster Yellow, it was "Geifuri 1". In Red/Blue/Yellow in America, his unused default name was "Ninten" (while the rival's was "Sony", going off the current console war at the time).
So yes, "Pokémon Trainer" is Red. He also isn't.
He's not defined by the NPC Red in Johto or Unova. He has a Squirtle and Ivysaur as opposed to their fully evolved forms.
He's defined by the Player in Kanto, who could have any set of Pokémon available to him within his player roles. Sakurai chose the iconic Kanto starter trio, but with different evolutionary levels for gameplay balance purposes. And since as a player, he does not have a defined name, his name is simply "Pokémon Trainer". Just like Villager.