In what sense do we say two functions are equal?

Suppose there are two functions $f: X to Y$ and $g: X to Y$. In what sense do we say $f = g$? In my understanding, the equality may imply
begin{equation}
left(forall x in Xright) left(fleft(xright) = gleft(xright)right),
end{equation}
or it may imply
begin{equation}
left(forall x,yright)left( langle x,yrangle in f longleftrightarrow langle x,yrangle in gright).
end{equation}
The former is the common sense for equality of two mappings while the latter is a general sense in set theory. In my opinion, the difference between these two understandings lies in how we interpret $X to Y$. If we deny $f$ or $g$ to be of elements $langle x,y rangle$ for $x notin X$ by using $f: X to Y$ or $g: X to Y$, then the two understandings are the same. Otherwise, there seems to be a confusion between common sense and what is defined in set theory. Probably a more consistent expression is
begin{equation}
left(forall x in Xright)left(fvert_{X} = gvert_{X}right).
end{equation}