Is there a socket adapter for the “cigarette lighter”?

2005 Prius, Gen 2.

I installed a CarPlay head unit, and unplugged and popped out the “cigarette lighter” 12v port in order to run its USB cable from the back of the stereo out of that hole in the dash. After that, I had the original cable and the plug that used to plug into the back of the 12v port.

I then realized I could get a 12V USB outlet and use that wire. However, the factory plug (that used to plug into the original “cigarette lighter”) has different positions for its + and - poles than the USB outlet I’m trying to add. (I’m sorry if these things have names that I don’t know).

1. is there some kind of adapter I could get so I could just plug into the outlet to the original plug, and not have to solder anything?
2. is there a way to “free” the wires from the plug so I could (say) solder spade connectors to the ends and slip them onto the new 12V USB outlet?
3. is the only way to get rid of the plug to cut the wires, strip them, solder to the wires that came with the outlet, and plug in?
4. the wires that came with the outlet have a fuse; do I need to retain that?

Thanks, hope this all makes sense. Pictures included

 

Thank you both for your suggestions. I ended up cutting the wire, connecting it to the wires that came with the 12V plug, and mounting it to the rounded dash part of the Prius. Photo attached. You'll notice the hole I drilled next to the previous location wouldn't fit the plug. I plugged that hole with a 3D printed "stopper"