The various versions of Lex Luthor.
PersonalityThis version of Lex Luthor is portrayed more as a mad scientist and power hungry dictator than the Lex Luthor we're used to seeing.
Lex is an incredibly skilled scientist, having used various inventions to conquer several minor European nations before being stopped by Superman and using the remainder of his resources to stop Superman.
This version of Lex Luthor became the ruler of numerous small European nations through his inventions before he was stopped by Superman.
Rivalry with Superman
Lex's rivalry with Superman stems from when Superman stopped him from causing a war within his small empire and has been his driving force throughout the rest of his appearances.
PersonalityThis version of Luthor is portrayed as a vengeful scientist who goes mad trying to prove he's better than Superman, starting from when they were teenagers and lost his hair. He's also much more maniacal than the previous Luthor.
Luthor is a very good scientist, having built various power suits to combat Superman and various other inventions. He also invents various other dimensions, even teaming up with Brainiac time to time. Being the greatest scientist is also one of his most important goals.
Luthor never becomes a president in this universe.