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.