Okay so... I can't even begin to describe the level of facepalm I am feeling...
It was the cable the whole time.
Just... why? I will never understand...
So, I have 3 devices:
The old phone (LG Vortex)
New phone (Galaxy S4)
Camera
and 2 cables:
Cable that came with camera (cable 1)
Cable that came with old phone (cable 2)
Cable 1 works with both the camera and new phone
Cable 2 works only with the old phone (I confirmed it does not work with both the new phone and this camera)
So, the old cable isn't broken. It's just... what? I don't even know what. Like they somehow made it so it will only work with my old phone? So here I was banging my head against the wall because I knew the cable still "worked", and it does still "work", but apparently only with my old phone.