Why did we call a row operation "elementary"? 
*

*Why we called the three actions of row operation "elementary"? Is there a thing called "advanced" or "complicated" row operation?

*I've seen the word "non-elementary" row operation is used to describe things like $R_1-R_2$, which is not written in the conventional $-1R_2 + R_1$. Is this usage correct? In particular, what should $R_1-R_2$ be called?
\begin{align*}
    &\text{a) A non-elementary row operation} \\
    &\text{b) An elementary row operation} \\
    &\text{c) Just a row operation} \\
    &\text{d) It is not a row operation}
\end{align*}
 A: The sense of "elementary" here is that all the operations that preserve the row-space of a matrix can be be produced by combining various elementary row operations.
Thus these are the elementary steps that can be taken to calculate a (reduced) row echelon form of a matrix, a basic tool for solving several kinds of problems involving the row-space of a matrix and the solutions (if any) of a linear system of equations.
With regard to what $R_1 - R_2$ ought to be called, something is missing from the description.  It would indeed be an elementary row operation if this "new row" immediately replaces $R_1$.  If you wanted it to replace $R_2$, you would have to perform that row operation by combining two "elementary row operation" steps (first replace $R_2$ with $R_2 - R_1$, then multiply the resulting new second row by non-zero scalar $-1$).
If you wanted to do something completely different with $R_1 - R_2$, then it would possibly either not be a row operation or not a row operation that preserved the row space of the matrix.  For example, if you replaced $R_3$ with $R_1 - R_2$ (leaving $R_1,R_2$ as they are), you might well be making the row space of the matrix smaller.
