I am trying to find the correct vocabulary to describe a distinction between two operations. The first is the simple translation operator, that when it acts upon a function $f(x)$ shifts the argument: $$ {\cal T}(a) f(x)=f(x+a) $$ This is a linear operator. The second might be called a ``shift" operator that acts upon a vector, $\vec x$. $$ {\cal S}(\vec b) \vec x = \vec x + \vec b. $$ This is distinct from the above because while $$ {\cal T}(a) (f(x)+g(x)) =f(x+a) + g(x+a) $$ The shift operator does not work that way: $$ {\cal S}(\vec b) \vec x + {\cal S}(\vec b) \vec y =( \vec x + \vec b) + (\vec y + \vec b) = \vec x + \vec y + 2 \vec b \neq (\vec x + \vec y + \vec b)= {\cal S}(\vec b)(\vec x+ \vec y). $$ Therefore this second operator is not a linear operator.
The problem is that many of my students have had both things called "translations" and superficially they look the same. I want to use the proper terminology (and perhaps clarifying terminology!) that will make the distinction clear.