This is not intended to be personal critisism of you, or any sort of attack on your credibility. I am merely trying to inject an element of scientific analysis and criteria, which is what is needed to actually answer the subject of this thread. Unfortunately, that means you have to be somewhat clinical in analysis, and whilst your efforts are appreciated, the gathered evidence isn't up to drawing much by way of conclusions.
1) Not quite sure what you're trying to say there... if it's that we have different opinions, then I dispute that slightly, although I realise some may consider it a pedantic point. We have different theories as to what makes a good flashlight for use in fog, but ultimately only valid and scientific experimentation will yeild a difinitive answer.
2) I never said tint was the only affect, but introducing such a massive flaw into the experiment from the outset just makes any conclusions you draw open to massive interpretation. I am well aware that CRI is also a factor, but a cool white tinted LED will have a massively different CRI to a warm white tinted one anyway, so that just adds more credibility to disputing the comparison. There's actually enough of a difference in your beam shape there to also affect clarity, but the most noticable factor is the beam tint - I would say the incan actually has a dark spot in the centre, which may possibly mean that the target is illuminated more from scattered light than it is direct light. If you want to draw a conclusion as to a particular parameter, you need to reduce all the other factors to virtually zero - your two shots have quite a few differing factors, so you can't conclude it is down to one factor. We already know tint as well as beam shape makes a large different, so both need to be eliminated to refine the tests and further our understanding. Another factor which unfortunately is beyond control is the density of the fog... is there a way to measure that?
3) We need proper comparisons as a frame of reference, yes. Photographs with the camera on the same settings for both are a good start, providing the light sources are comparable. I have already stated why, in this case, the comparison lights aren't close enough in real world output to draw much of a conclusion.
4) I don't have enough light sources to do many comparisons, but I do have a camera with proper colour calibration setup from an industry recognised Kodac colour chart at all ISO and Shutter combinations with the different lenses I use. I also have access to a 'dark site' where there is virtually no light pollution present. I would be more than happy to do beamshots in cooperation with somebody who could provide flashlights. The biggest issue I would have is actually that we get very little fog in controlled conditions - it tends to be somewhat unpredictable! There seems little point in me posting beamshots of a common flashlight in commonly occuring conditions on this forum, there are already plenty if you use a search.