Clearly we have differences in what is intuitive and what is not. I think OP is trying to say that people who interpret Case 1 and Case 2 in terms of absolute

Clearly we have differences in what is intuitive and what is not. I think OP is trying to say that people who interpret Case 1 and Case 2 in terms of absolute difference in number of heads rather than the ratio of heads to tails have questionable foundations in probability. I probably won't be able to change your intuition, but let me try to show why thinking in terms of difference in log-odds is grounded on nice mathematical properties. When thinking about how to compare two probabilities (i.e. a function that takes two numbers between 0 and 1, and outputting a real number), one thing that is nice to have is the function outputting opposite numbers for the complementary event. A comparison function that fulfills this property (and other nice properties) is the difference in the log-odds. For example, using this comparison for p1=0.002, p2=0.001 gives about 0.694. What's nice is that when p1=(1-0.002) and p2=(1-0.001) you get -0.694. So indeed this comparison function is giving expected results. Using this comparison function to look at 0.500 and 0.499 gives 0.004, which is less than 0.694. This suggests that using this mathematical framework to compare probablities, (0.500, 0.499) is similar while (0.002, 0.001) is more different. Indeed if you take the extreme case of 0.001 and 0 (I made a typo above) you see that the comparison function outputs infinity. This also makes sense when making decisions because when comparing an event that is impossible compared to an event that is merely improbable, the actual probability no longer matters for the latter event (think of your coin flipping case again with p1=0, and letting the other p2 be any number between 0 and 1).