How do you differentiate between knowledge and belief?

As neat and tidy as your explanation is, I think you are vastly oversimplifying the concept.

You say the moon is real because you can see it, and you can prove it's there by telling other people to just go look at it. Alrighty then, I've seen bigfoot. In fact, lots of people say they've seen bigfoot. Therefore he must exist too, right? The photos "prove" his existence just as much as you pointing to the sky saying the moon exists cause there it is.

Now, I realize that there's probably some degree of hyperbole in your statement, so I'll walk this back a little. If the defining metric of your separation between these concepts is whether the hypothesis can be proven through experimentation, that's all well and good. However, I would argue that, in 99.9% of cases, it's still a belief statement. Let's continue with the moon example, but, rather than "seeing is knowing", let's apply the same standard that you applied to God. So, you "know" the moon exists, not just because you can see it, but because it's existence can be empirically proven through experimentation. What sort of experiments would you conduct to do that, exactly? Have you done those experiments? Or, like the rest of the rational world, do you accept that scientists have done those experiments already and decided, yup, moon's there? Cause, if you're taking someone else's word for it, do you personally "know" what they are saying is true, or do you believe them based upon their credentials, the credentials of those who support the argument, and your own personal beliefs/knowledge?

As another example, let's imagine for a sec we're philosophers/scientists of the ancient world. I have a theory that the heavier something is, the faster it will fall. You may know where I'm going with this if you remember your elementary school science classes. I believe in the power of experimental evidence, and so, to test my theory, I climb to the top of the Acropolis and drop a feather and a rock. The feather falls much more slowly than the rock. Eureka, I've proved my theory and therefore I now KNOW that an object's weight affects its fall.

Now, anyone not born in 850 BC Athens in this thread will point out that it's a flawed experiment, since I'm not controlling for air resistance, and if you conducted the same experiment in a vacuum chamber, both objects would fall at the the same rate. However, the technology to test my hypothesis with all of the salient variables controlled did not exist at that time. So, even though it's now widely known that my experiment was flawed, it wouldn't have been at the time, and I would have the data to back up my theory. I could simply say try it yourself, it's a self-evident fact.

Finally, your statement about subjectivity of definition being an obstacle to functional language is so alarmist as to border on ridiculous. If this question were "how do you personally define the distinction between 'yes' and 'no'", then sure I can get on board a little bit more with your point. However this is much more like 'twilight' vs 'dusk'. Crack open a dictionary and you'll find that there is a stark, objective distinction between those terms, much as you pointed out that belief and knowledge have very different definitions. For the record, since I had to look it up to ensure I wasn't telling tales here, sunset is the moment the sun finishes crossing the horizon, twilight is the period between sunset and dusk when light is still in the sky but the sun is not, and dusk is the moment the sun is 18 degrees below the horizon. So, I know that these are unique terms with specific, mutually exclusive definitions. But let me tell you something, I believe that if I randomly substituted one term for another based purely on my personal whimsy, people are gonna get what I mean regardless.

That's incredibly naive I'm afraid. This sort of logic works in very simple cases, but quickly breaks down in any complex scenario. The reality is that a lot of knowledge cannot be easily verified because it's just too complex. Take a peer reviewed scientific study as an example, the study might reference a different study as its basis, that references another study, and so on. If one of the studies in the chain wasn't conducted properly, and nobody noticed then the whole basis could be flawed. This sort of thing happens all the time in practice.

What you really have is an ideology, which is a set of beliefs that fit together and create a coherent narrative of how the world works. A lot of the knowledge that you integrate into your world view has various biases and interpretations associated with it. Thus, it's not an absolute truth about the world, but merely an interpretation of it.