Your original arrangement and code appears to be quite correct. Don't bother about changing the pin numbers.
You have the contrast set more-or-less correctly, but the display is not being initialised. Something must be wrong with the connections.
As you have no LED to illuminate this display, you do not need to connect pins 15 and 16, so you may as well remove the connections to those so as not to confuse yourself.
I suggest you take down all the jumper wires, test all of those with a pin on each end using a LED and 330 ohm resistor on the breadboard which you connect using each of the jumpers (two at a time) and if they are all correct, carefully reassemble it according to your original diagram.
Actually, the jumper wires you already have connected to Vcc and ground and the potentiometer are clearly OK as they are - but I have to suspect there is either a faulty wire, a wire not properly inserted, or a wire to the wrong pin that I just cannot see in your photograph.