How come people write the integers mod $n$, i.e. the quotient group $\mathbb{Z}/ n\mathbb{Z} \equiv \mathbb{Z}_n$ as
$\mathbb{Z}/ n\mathbb{Z} \equiv \mathbb{Z}_n = \{0,1,2,...n-1\}$?
Isn't this incorrect? Should it not be $\mathbb{Z}/ n\mathbb{Z} \equiv \mathbb{Z}_n = \{[0],[1],...,[n-1]\}$ where $[a]= a + n\mathbb{Z} = \{b \in \mathbb{Z}: b \equiv a$ mod $n \}$ are the cosets. I am quite confused about this. Clearly the elements are not the same.
As discussed in the comments, saying "$\mathbb Z/n\mathbb Z=\{0,1,...,n-1\}$" is not formally correct but more of an abuse of notation.
Something that can be done to somehow justify this, is defining a system of presentatives for the quotient group as follows:
Let $X\subset G$ we say $X$ is a system of representatives of $G/H$ if:$$|X\cap gH|=1 ,\forall g\in G$$ Having defined that, one can see the map: $$X \longrightarrow G/H \\ x \mapsto xH$$ is a bijection from the system of representatives to the quotient subgroup. So in your particular example saying "$\mathbb Z/n\mathbb Z=\{0,1,...,n-1\}$" is a way of saying the set $\{0,1,...,n-1\}$ is a set of representatives of $\mathbb Z/n \mathbb Z$.
I hope this helps.