Difference in definition of a cover Let $X$ be a topological space,

Ive seen some define a cover, as a collection, $A$, of sets such that $X=\bigcup A$, while others use the condition $X\subseteq \bigcup A$.

Why the difference? My lecturer uses the first definition but to me the second seems more natural. After all, to cover something doesn't necessarily mean to cover it exactly.
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.
$X\subset\bigcup A\implies X=\bigcup \left(X\cap A\right)$