Are multivariate and cumulative exchangable in copula? In copula model, some researchers, identify it as a multivariate distribution function, while other present it as a cumulative distribution function. I believe multivariate differs of cumulative. But it seems researcher used them as exchangeable definition. Is that correct?
My other point is, copula is known to be with standard uniform margins, so why author used the following form of definition:
Copula is a multivariate distribution function with standard uniform margins U(0,1) on [0,1].
So, my point is, why they say U(0,1) on [0,1]. What is the difference between U(0,1) and on [0,1]?

The two terms are not synonyms.
Copulas are necessarily multivariate.
The conventional definition of a copula function is of $C$, a cdf but someone might occasionally refer to the density $c$ as a copula since it contains the same information. I'd be more likely to say the density of the copula but not everyone does, particularly in informal discussion.
So $C$ will be a multivariate cumulative distribution function.
