I have used several different Firefox extensions that will accomplish your goals. The best one I have found is Update Scanner by sneakypete81. It is available here:
It is gratis (free of charge), and like all extensions, is open-source. I never had any issues with false positives.
However good it was, it was not flexible enough for my use. I continued to search for tools, and I eventually found WebChangeMonitor. It's not a web browser extension, but it is so much better than all the web browser extensions I tried. It is a portable standalone software for Linux and Windows. It is available here:
WebChangeMonitor is also gratis and open-source. It is very flexible, and will do exactly what you want. You can configure it to eliminate just about any false positive as well as detect changes in a specified section of any page. I highly recommend it.