Confusion over the usage of different terms for ref and rref. My school uses row echelon form and reduced row echelon form to denote their respective types of matrices which results from gauss jordan elimination.
Today I came across this term used in this sentence.

The nonzero rows of a row reduced echelon matrix form a basis for the row space of the matrix.
The nonzero rows of a row reduced echelon matrix are independent.

http://www.millersville.edu/~bikenaga/linear-algebra/rank/rank.html
I am confused about the term "row reduced echelon". Which is it referring to?
Also I would like to clarify if,

The nonzero rows of a Row Echelon Form matrix form a basis for the row space of the matrix. (i.e non RREF)

 A: Row reduced echelon is the same thing as reduced row echelon form (although admittedly, the terminology is a bit weird). The rows of the row echelon form does indeed form a basis for the row space.
A: Let's take an example.  Consider the matrix
$$
\begin{bmatrix}
1 & 5 & 3 & 2 \\
-4 & -2 & 1 & 5 \\
0 & -1 & -1 & 2
\end{bmatrix}
$$
Step 1:
$$
\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 18 & 13 & 13 \\
0 & -1 & -1 & 2
\end{bmatrix}
$$
Step 2:
\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 18 & 13 & 13 \\
0 & 0 & -5/18 & 49/18
\end{bmatrix}
Step 3:
\begin{bmatrix}
1 & 5 & 3 & 2 \\
0 & 1 & 13/18 & 13/18 \\
0 & 0 & 1 & -49/5
\end{bmatrix}
I make no claim that terminology regarding this is standard, but this is what I consider to be row echelon form (REF).  That is, you reduce the matrix until you what you get is `upper-triangular'.
Step 4:
\begin{bmatrix}
1 & 5 & 0 & 157/5 \\
0 & 1 & 0 & 39/5 \\
0 & 0 & 1 & -49/5
\end{bmatrix}
Step 5:
\begin{bmatrix}
1 & 0 & 0 & -38/5 \\
0 & 1 & 0 & 39/5 \\
0 & 0 & 1 & -49/5
\end{bmatrix}
On the other hand, this is what I regard as row-reduced echelon form (RREF).  That is, after your matrix is in REF, you continue to row-reduce `upwards' until there is nothing left to do.  I realize these definitions are imprecise, but hopefully it should now be clear what is meant by "row echelon form" and "row-reduced echelon form".
The last statement you made is true (regarding the row space of the matrix).  You should check yourself that each 'rule' of Gaussian elimination does not change the row space of a matrix, hence, it doesn't matter if you row reduce to REF or all the way to RREF:  if each step doesn't change the row space, then no matter how you row reduce, the row space is going to stay the same.
