I understand that the solution space of the augmented matrix $AM =$ $$ \left[\begin{array}{ccc|c} 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \end{array}\right] $$ is considered to be a "$1$-flat" in $3$-space, because there is one "free" variable, but what if I have $$ \left[\begin{array}{cccc|c} 0 & 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & 1 & 2 \\ \end{array}\right] $$ as my matrix in $4$-space? Would it be called a "$2$-flat", because there are two independent variables? Is there any nomenclature distinction between the variables $w$ and $z$ in the variables vector $X =$ $$ \begin{bmatrix} w \\ x \\ y \\ z \\ \end{bmatrix} $$ where $AX = B$, $B =$ $$ \begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ ?} $$
The $w$ and $z$ variables are both independent, but $x$ and $y$ are only dependent on $z$, not $w$.
It is a $2$-flat -- you're correct. And I don't know of any nomenclature distinction between free variables that other variables depend on and ones that other variables don't depend on. Both are just free (or independent) variables.