Everybody knows the definition of "being continuous on a point" and it's defined well.
But the definition "A function $f$ is said to be 'continuous' if it is continuous at every point of its domain" is not adequate I think.
For example, let $f(x) = \frac{1}{x}$.
This function is continuous on $(-\infty,0)\cup(0,\infty)$ which is also its domain. Then, is this function continuous?
It bothers me to claim that this function is continuous. Why would we define such a function as continuous? What happened to "drawing the graph without raising your hand"?
The primitive notion of continuity is that which only concerns a single point. We then can say "continuous over/on $S$" to mean "continuous for all points in $S$"*
What "continuous" means without qualification is ambiguous, although usually clear from context. Often times it will mean "continuous on its domain" simply because that's a sufficient analytical property to consider a function "nicely behaved" in order to prove further theorems.
What you've noted is that in the secondary school context, "continuous" does not mean "continuous on its domain," because mathematics at this level has yet to fully distinguish the descriptive and computational interpretations, and endeavors to retain the idea that you can plug a real number into a function even if that number is not in its domain (though of course, not necessarily resulting in any sense). As a reflection of this, in this context "continuous" means "continuous on the reals." $1/x$ is not continuous on $\mathbb{R}$, in line with this intuition.
*NB: formally, for points on the boundary of a set a function need only be continuous from the interior