I think in any definition of science it is of paramount importance to be clear, that science makes no claims to absolute truth or reality
I don't think a short definition of science needs to say this but a good discussion of science certainly should. The definition I'm working on will go in a video that will break down and expand the definition for the viewer. It's in that breakdown that I'd talk more about things like "truth" and "reality".
The definition I originally used in my debate with Steve was something like this: "Science can be thought of as humankind's ongoing, collective, systematic attempt to accurately describe observable reality". I'm no longer happy with that for several reason, one is that the word "reality" implies that we can use science to come to absolute knowledge about reality. While this would be nice and might be the ultimate goal of some researchers, it may be impossible and is certainly not the immediate goal of scientific research.
if there is one simple fact that is true: things exist which science can never measure.
I certainly agree that science is limited in the kinds of things it can study but I may be more optimistic than you about it's ability to gradually overcome many of those limitations. As technology increases and our reach of our senses expand, our ability to observe and measure things also increases.
Steve and I have talked a bit on the radio about natural vs supernatural. I'm currently under the persuasion that things we call supernatural are simply things which we have have not yet been able to observe, explain, or measure. If God exists, for example, then all things from his perspective, including himself I assume, would be natural and within the realm of his scientific exploration (not that he would need to explore but I think you know what I'm saying). This may just be a matter of semantics but I don't think it is. I think if God exists, he is Natural, even if he inhabits a part of nature we may never get to experience.
If we prematurely say "That is outside the realm of science" we miss out on potential discoveries. This is why I don't like typical ID or Creationist claims. What I hear them saying is essentially this: "God did, we need not explore further". Ken Ham is notorious for saying this sort of thing. Micheal Behe and the ID crowd do it as well but with more sophisticated language. Even if it's true that God did it, a good scientist would say "okay, maybe, but how?".
In short, when confronted with what seems to be an impossible question, I think it's better to say, "that currently seems to be outside the realm of science" instead of saying "that is outside the realm of science".
In the light of that here is my attempt (avoiding the words "fact" and "verifiable," words charged with too much bias):
I think you may be right about the word "fact" being too charged. "Data" might be better. What do you think?
Building a contextual framework from the unproven and assumed foundations of cognitive awareness and logical thinking
I see what you're saying here. I think my definition, though it does not say this explicitly, assumes it or makes room for it by saying that science is "an ongoing conversation about how those facts can be best linked together". I agree with you that "The unproven and assumed foundations" of logical thought are worthy of discussion.
organize and understand all that can be observed or thought of with physical senses
I don't know what you mean when you say all that can be "thought of". Please explain.
in a cooperative community
I used to have something like this in my definition as well but I scrapped it because I think science can be done personally as well. In other words, it's nice to share what we find, but we don't have to.
with the understanding that any currently held idea might be superseded by something unknown; and the constant willingness to re-evaluate and reassess all current arguments and data, to better understand them;
Again, I think I cover this sufficiently in my definition but it is worth lots of discussion in the rest of the video.
while attempting to be as rigorous and disciplined as possible.
This is another really good discussion point. Not all facts are documented equally, nor are they equally verifiable. Some of this is due to our current technological limitations or unfortunate circumstances. Some is due to poor research standards. Fossils are notoriously difficult to study because they happen to be valuable and highly guarded. Besides just limiting who gets to see and handle a fossil (which limits the number of people who get to critique the conclusions of paleontologists), the money involved can also incentivize fraudulent "discoveries".