Schervish's (1995) Theory of Statistics defines exchangeability like this (p. 7):
A finite set $X_1, …, X_n$ of random quantities is said to be exchangeable if every permutation of $(X_1, …, X_n)$ has the same joint distribution as every other permutation. An infinite collection is exchangeable if every finite subcollection is exchangeable.
Likewise, Wikipedia currently says:
Formally, an exchangeable sequence of random variables is a finite or infinite sequence $X_1, X_2, X_3, …$ of random variables such that for any finite permutation of the indices, the joint probability distribution of the permuted sequence is the same as the joint probability distribution of the original sequence.
In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d.) if each random variable has the same probability distribution as the others and all are mutually independent.
But what exactly is meant in these cases to say that, e.g., a random variable $X$ has "the same distribution" as another random variable $Y$? Does it mean that the CDFs of $X$ and $Y$ are equal? Or does it mean the CDFs are equal almost everywhere? Or is it something else?