Completeness is a property of a family of probability distributions, not of a particular distribution. In the textbook, I see that a binomial family is a complete statistic. Here, the object of completeness is a family distributions. But I also see that given a iid uniform $(0, \theta)$ observations, $T(X)=\max_i X_i$ is a complete statistic. This goes back to my familiar feeling, such as a complete statistic could be max, min, sum, order statistic, etc in the section of sufficient statistics.
However, in the first example, we directly say a binomial family is a complete statistic.
They make me very confusing.
Saying 'a binomial family is a complete statistic' makes no sense.
What we can say is that a binomial family of distributions $\{\text{Binomial}(n,\theta):\theta \in [0,1]\}$ is complete. Here the family of distributions is indexed by the parameter $\theta$.
This means that if $X\sim \text{Binomial}(n,\theta)$, then $\operatorname E_{\theta}[g(X)]=0$ implies $g(X)=0$ almost surely for any (measurable) function $g$.