I have the following definitions in my notes for arbitrary discs and open balls -
$$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$
$$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$
The $\mathbb{R^{n+1}}$ seems wrong to me...Earlier in my notes I have that $D^2$ is a disc in the plane? So which is correct?
I like to think of that $n$ as the dimension of the underlying manifold (I guess this is also the reason behind this). So you can remember, that $S^n$ is defined as the unit vectors in $\mathbb{R}^{n+1}$, since it is an $n$-dimensional "object". This is also, since it is the boundary of $D^{n+1}$ which is $(n+1)$-dimensional.
Therefore I would definitely say $D^n,B^n \subset \mathbb{R}^n$. This is standard, since they are $n$-dimensional. But be careful: while the dimension is standard, different authors use $D$ and $B$ for different purposes, so $B$ happens to be closed from time to time.