In the usual sense, given two spaces $X,Y$ and a family of functions $F$ which map $X$ into $Y$, we say $F$ separates points if for any distinct $x_{1,2}\in X$ there exists at least one $f\in F$ such that $f(x_1)\ne f(x_2)$.
Now comes a separating family of seminorms defined on a vector space. However, in this case the definition of "separating" doesn't seem to align with the usual definition, because for a seminorm family $P$ to be separating, we only require that for any $x\ne 0$, there exists $p\in P$ such that $p(x)\ne 0$. But $P$ may well not separate points. For example, just let $P$ consist of the norm on any Euclidean space, then $P$ can't separate any sphere.
My problem is a rather soft one: why did mathematicians not avoid this confusion when they created the terminology "separating family of seminorms"? Is the reason purely historical (like, the two terminologies were created concurrently and independently)? Or any deep mathematical reasons?
The separating semi-norms separate points: For distinct points $x_1,x_2$, there is a semi-norm $p$, such that $p(x_1-x_2)\ne0$.