I found the problem!
You're moving only the player's sprite. You need to move the whole player node.
In the script Area2D_player.gd, change:
onready var player = $Sprite_my_player
to:
onready var player = self
Or a cleaner solution is to remove the "player" variable declaration and all its usages.
For example, change:
player.position.x += player_speed * delta
to:
position.x += player_speed * delta
A Godot debugging setting that would have helped solve this is:
Debug >> Visible Collision Shapes
With that setting enabled, in the original code, it's clear that the player's sprite moves but the player's collision shape stays in one place.
