*Puts on nitpick glasses* Erm, actually, the Elite Four's origina Japanese name is "
The Four Heavenly Kings", named after and based on the concept of four Buddhist Gods whom watch the world according to one cardinal directions (north, south, east, & west). *Takes off glasses*
As for the Champion, there's really no definite thing which makes up the Champion aside they're a strong trainers. Using the criteria you listed above:
Blue: fossil,
pseudo,
evolution gimmick (Alakazam, Exeggutor, Arcanine),
obtain gimmick (Starter)
Lance: fossil,
pseudo (multiple of Dragonite),
evolution gimmick,
obtain gimmick (Charizard)
Red: fossil,
pseudo,
evolution gimmick (Espeon in Gen II),
obtain gimmick (all 3 Kanto Starters, Snorlax, Lapras in Gen IV)
Steven: fossil (both gen's),
pseudo (though only one of his gen's),
evolution gimmick,
obtain gimmick (not counting pseudo)
Wallace: fossil,
pseudo,
evolution gimmick (Milotic, Ludicolo),
obtain gimmick (sorta, Wailord and Whiscash needed for puzzle to solve Regi Puzzle)
Cynthia: fossil,
pseudo,
evolution gimmick (Milotic, Togekiss, Roserade, Gastrodon to an extent),
obtain gimmick (Spiritomb)
Alder: fossil,
pseudo,
evolution gimmick (Accelgor & Escavalier),
obtain gimmick (Volcarona)
Iris: fossil,
pseudo (though not main Pokemon),
evolution gimmick,
obtain gimmick
Diantha: fossil (both gen's),
pseudo,
evolution gimmick (not counting fossil/pseudo/Mega),
obtain gimmick (Gourgeist to an extent)
And that's not counting other things like how the Elite Four sometimes picks up the slack with some owning the pseudo if the Champion doesn't (Lance in Gen I, Drake uses a Salamence in Gen III).
Wonder if this may also be true for the Ultra Beasts, having a code name now but a name later on.