The difference is : is $X$ a subspace of a larger space $Y$ ?
If it isn't, you only have one natural definition : a cover is a collection of subspaces $A_i$ such that $X = \bigcup_i A_i$.
But if $X\subset Y$, then you have two natural definitions : a cover of $X$ can either be a collection of subspaces of $X$ such that $X = \bigcup_i A_i$ (def $1$), or a collection of subspaces of $Y$ such that $X\subset \bigcup_i A_i$ (def $2$).
Of course a cover for def $1$ is in particular a cover for def $2$. And if you have a cover $(A_i)$ for def $2$, you get a cover for def $1$ by using $B_i = A_i\cap X$.
This behaves well because for instance $B_i$ is open/closed in $X$ iff $A_i$ is open/closed in $Y$. On the other hand, if $(A_i)$ is an open cover for def $1$, it may not be an open cover for def $2$, because an open subset of $X$ is not necessarily open in $Y$ if $X$ is not itself open if $Y$. So in that case you have to replace $A_i$ with $A'_i\subset Y$ such that $A'_i$ is open in $Y$ and $A_i = X\cap A'_i$ (which exists by definition).
In the end you can see that the two definitions give the same thing if you're ready to modify a little your covers when you switch definitions.