A first broad attempt.
The stage is set-up in this way: you observe two set of data, that your model indicates as coming from two distinct sources. The question is: are the two sets related in any way? If so, how much? The 'measure' of such is usually called correlation.
From an objective Bayesian point of view, it doesn't make much sense to talk about correlation between two random variables (it makes no sense to talk about random variable either, but that's another story), because correlation is always model dependent, and probabilities are epistemic. Two agents observing the same phoenomenon, having different information about it, may very well come to totally opposite conclusions.
From a frequentist point of view, though, the correlation between variables express an objective quantity, all the measure that you mention are attempts at finding out how much correlation there is, making more or less explicit assumptions about your model.
If you think that the two sources are linearly related, then the Pearson coefficient will tell you how much the data supports the model.
If you think the two variables comes from a continuous normal distribution, but you can only observe their integer value, you use polychoric correlation. And so on...
Depending on the assumptions you make, there are different measures of how much correlated the data are.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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