Why is not the definition of probability distribution consistent with the definition of Binomial Distributions?

Definition of Probability Distribution:

A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence.

And,

Definition of Binomial Probability Distribution:

A binomial distribution is a specific probability distribution which is used to model the probability of obtaining one of two outcomes, a certain number of times (k), out of fixed number of trials (N) of a discrete random event.

Are there two definitions consistent with each other? I think no.

Why? Because, how is a Binomial Distribution acting like a table or equation that links each out come of the experiment with its probability of occurrence?

• they are perfectly consistent. the binomial probability distribution gives you the probability of the number of successes in a set of n independent yes/no experiments, each one having identical probability of success p – matteo Oct 4 '16 at 20:51

Imagine an experiment where you toss a coin $N$ times and count the number of heads that occurred. Imagine that the coin doesn't necessarily have to be fair and it can return heads with some probability $p$. Binomial distribution is a function parameterized by $p$ and $N$ that maps the number of heads that occurred, $k$, with probabilities. In here $p$ and $N$ are features of the experiment and $k$ is the outcome. So yes, it's consistent with the definition provided.