Sam is correct. The color of the connectors makes no difference - the colors are there to help you make sure both ends of the same physical wire are connected correctly.
1) For "regular" karaoke, as Sam suggested, use either the coaxial or optical output on the karaoke machine to the corresponding digital input on the AVR (the receiver). Since you have lots of coaxial cables, it probably makes sense to just use one of those for that connection. That's all you need for "regular" karaoke.
2) For "pc karaoke", the only audio FROM the karaoke machine that you want is the microphone audio. HOPEFULLY, that is available via the center channel output on the karaoke machine - again, we don't know for certain. Pick an RCA cable and connect one end to the "center" output on the karaoke machine (like you have done) and the other end to the "center" input on the back of the receiver.
3) For the music part of "pc karaoke", it sounds like you have that set up correctly - using the headphone output from your PC to the Left/Right multichannel inputs. What this means is that when you select the multichannel audio input on the AVR, you will get Left/Right from your PC and center channel from the karaoke machine.
4) We also forgot about the video signal from the karaoke machine. Pick another RCA cable and connect it from the yellow "monitor out" on the karaoke machine to the yellow "in" on the AVRs DVD input. This will work for both forms of karaoke - any time you switch to the DVD input on the AVR, it will show the video from the karaoke machine.
With both devices turned on, select the DVD input on the AVR. If the DVD input is already assigned to the digital coaxial input we used in step 1, regular karaoke should work. If not, you may need to go to the input assignment setup menu on the AVR to make sure that when you switch to DVD, the coaxial digital audio input is used.
For pc karaoke, you will once again select the DVD input (with both devices turned on), but this time, you want to use the multichannel input (instead of the digital coaxial). According to Sam, that should mean pressing the "Multi-Ch" or some such button on the remote control. Consult your manual if you're not sure.
Here's a slightly edited version of your picture, with some connection guidelines.
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