I am learning ideal arithmetics and I was flabbergasted that $\langle x\rangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read this paper:
A binomial is a polynomial with at most two terms; a binomial ideal is an ideal generated by binomials.
So in the context of Algebraic geometry, how can I differentiate the two different meanings about binomial? When does "binomial" mean two and when does "binomial" mean at most two? Why is such difference made?