the distribution as the complete probabilistic description

The distribution of a random variable is the full answer to every question of the form how likely is X to land in this set of values? It is not a single number like a mean; it is the entire pattern, the complete list of how probability is allocated across all possible values. Once you know the distribution, you know everything there is to know about X from a probabilistic standpoint, and any specific quantity (its mean, its variance, the chance it exceeds 10) is just one feature read off from it.

You can present the same distribution in several equivalent ways, and they all carry identical information: the pmf for a discrete variable, the pdf for a continuous one, the cdf for any variable, the survival function, or the quantile function. Pick whichever is convenient; they are interchangeable views of one object. The practical meaning is sharp: if two random variables have the same distribution, then every probability statement about one is also true of the other, even if they arise from totally unrelated experiments.

This is why we so often say a variable is, for example, normal or Poisson, naming its distribution rather than the experiment behind it. The experiment is scaffolding; the distribution is what survives and what theorems are about. A common confusion is to mistake a summary for the whole: the mean and variance are useful features, but two very different distributions can share both, so knowing a couple of summaries is not the same as knowing the distribution. The distribution is the object; summaries are shadows of it.