Why did we call a row operation "elementary"?

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.