My definition of a site is a pair $(\mathcal{C},\DeclareMathOperator{\Cov}{Cov}\Cov (\mathcal{C}))$ where $\mathcal{C}$ is a category and $\Cov (\mathcal{C})$ is a **set** consisting of elements of the form $\{U_i\rightarrow U\}_{i\in I}$, and $\Cov (\mathcal{C})$ satisfies three axioms which I gonna omit since this is not the problem.

I have searched a few books about the definition of an big/small étale/fppf/... site.

For example, a small étale site of a scheme $X$, $X_{ét}$, has the underlying category whose objects are étale morphisms $U\rightarrow X$, and the covering set $\Cov (X_{ét})$ consists of all surjective families of étale morphisms $\{f_i:U_i\rightarrow U\}_{i\in I}$, i.e. $f_i$ is étale and $U=\cup_i f_i(U_i)$.

But those books don't mention that why the "covering set" is a set.

"Introduction to étale cohomology" by Gunter Tamme, Manfred Kolster did that.

"Étale Cohomology" by James Milne did not mention the definition of an arbitrary site, he just define a big/small $E$-site for a class of $E$-morphisms of schemes.

"Lecture notes on étale cohomology" by James Milne on the website see this link, he defines an arbitrary site s.t. for each object $U$, we have a distinguished set of covering families, and this is a system of coverings, didn't say the whole thing is a set.

Stacks project, uses a confusing way to construct the underlying scheme of a small/big étale/fppf/... site, so I have to ignore it.

SGA is in French which I have an incredibly slow reading speed, so ignored.

Which definition of site (arbitrarily) and small/big étale/fppf/... site should I accept?