What is the interior of the standard-$n$-simplex?

Taken from here Boundary/interior of $0$-simplex

The standard-$n$-simplex $\Delta^n$ is the subspace $$ \textstyle \Delta^n = \{x=(x_0,\dots,x_n)\in\Bbb R^{n+1}\mid \sum_0^n x_i=1,\,x_i\ge0\,\forall i \} $$

Question what is the interior of the above simplex?

Originally, I thought it is simply:

$$ \textstyle \text{int}\Delta^n = \{x=(x_0,\dots,x_n)\in\Bbb R^{n+1}\mid \sum_0^n x_i<1,\,x_i\ge0\,\forall i \} $$

But then it seems stuff in the interior do add up to $1$.

So instead it should be:

$$ \textstyle \text{int}\Delta^n = \{x=(x_0,\dots,x_n)\in\Bbb R^{n+1}\mid \sum_0^n x_i=1,\,x_i>0\,\forall i \} $$

Which one is correct?

The second is correct. Let $n=2$. Your standard simplex is then $x+y+z=1, x,y,z \ge 0$. The boundary has at least one coordinate equal to zero, two in the corners. The interior then has all coordinates positive.