Sorry. That was confusing what I wrote.
Correct, for directional light the position doesn't matter. I meant that the light is in a strange angle (lighting the back of the character) so that is why it looks off from the reference. You should try rotating the light so that it is facing down and more to the side of the character (not facing the back).
Probably writing your own shader is not needed. Though it may give you a more custom look, if you are not familiar with it, I'm sure there is a way to tweak the settings in Godot to get something better.
You might want to play with the environment node a bit. You can set ambient light strength and color, sky energy, and stuff like that so your scene isn't so dark. Upping the energy on the directional light should also make it look less washed out.
Sorry for the confusion.