Arduino is connected to PC via standard USB cable and that's how it is powered. The LED strip is powered from the PSU.
Yes, the connector going into Arduino's GND pin between AREF and pin 13 is the one that melted.
That connector is connected to the LED strip's GND which is also connected to the PSU's GND (actually COM is the label).
In the photo I disconnected the Arudino from USB (forgot to reconnect it for the photo), but other than that, that's exactly the connections I have - 2 jumper cables (GND that I already talked about and pin 6 that is connected to the LED strip's DIN) plus the USB cable that is the standard cable that came with the Arudino and it's connected to my PC's USB port and is providing the power for the Arduino itself, while the LED strip is connected to the external PSU. Because they share a common ground I imagine that's the issue, but I still don't know what to do about that.
Also, I'm not saying it's 100% correct, but it has been working connected like that for the past almost a year, it's not a new connection that immediately went wrong.