I will first admit that I am not an expert in this subject. However I have had some trouble and worked though it in the past. That is just to say my help may not be enough to get you to a solution. 
Your settings in the plugin look good. The plug in I think is mostly about how to show the camera inside OctoPrint and does not really do much related to the camera streamer itself.
When it "does not work" what do you get when you push the test buttons?
What do you get when you try to connect directly to the stream.
URL should be something like this: (replace the "192.168.x" with your IP)
http:\192.168.0.x\webcam?action=stream
If you cant get your stream there, it is likely an issue with the config of your USB Camera in the Config.txt related to webcamd.
Looking at your "webcamd.log" file, It looks like it is running as expected but you might need to update the /boot/config.txt to list your specific USB device. This is noted in the log. It might be just fine to have this but when I look at one of my systems that is running the same stack, my log is more or less empty. So I am thinking there might be something to this.
To learn about the webcamd streamer, you could take a look at the man page for it here:
It might give you some insight in to what your issue is.
Also note that you are looking at the stream through routing set up in the HaProxy. HAP is using the path in the URL to direct the requests to the proper 8080 local port. I don't expect anything is wrong here or you would not have seen anything when you said you had it.
For me I would look into these possible causes as suspect.
Your service is failing to start on reboot. (Low likely given the webcamd log file
The service for the webcamd can be manipulated with commands like:
sudo systemctl restart webcamd.service
sudo systemctl start webcamd.service
sudo systemctl stop webcamd.service
sudo systemctl status webcamd.service
or
You need to configure the proper USB camera in the config.txt
Dig in a bit and see what the status of your service is as well as what happens when you use the test buttons and directly access the stream URL. Form there you may workout the issue yourself.
