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Confused about c-betting strategy IP

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Confused about c-betting strategy IP

Specifically I was looking at button vs. BB compared to UTG vs. BB. I'm a little confused about why these strategies change the way they do depending on the different ranges.

My understanding so far (please correct me if wrong) is that:

On a dry board like A22, K27 etc. we want to be betting very often both as button and UTG because we have range advantage. As button we bet smaller because while we have nut advantage, it's less pronounced given ranges are so wide. As UTG we can choose a bigger sizing because nut advantage is more pronounced given our range is so much more condensed.

What I'm confused about is on a wet board e.g. 567 twotone. I've heard that as button we are betting polarized, choosing a bigger sizing to get value, protection, fold equity etc. from our nutted hands and high equity bluffs. In Pete Clarke's video UTG vs. BB he goes over the same spot as UTG in pio, and it's mixing between a bigger and smaller sizing. He recommends just simplifying and using the smaller bet as both our range advantage and nut advantage is reduced, thus we have to bet infrequently and with a smaller sizing. Both explanations make a lot of sense to me, but I'm confused as to why these strategies differ in these positions. Would our range advantage and nut advantage not be quite similar both UTG and button given that the board is so equalising? Why are we betting a bigger size when polarizing on the button, and a smaller size UTG?

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