Quoting from the question:
$F(aK,aL)$ : simultaneous and proportional inputs increase
$aF(K,L)$ : output increase
But $F(aK,aL)$ is an output increase, that is why it starts with $F$. It is the output increase due to the simultaneous and proportional increase of inputs. At the same time, $aF(K,L)$ is a hypothetical output, the output were this if ...
Returns to scale is directly related to homogeneous functions:
For a homogeneous function $F(x,y)$ given $\theta>0$, for simplicity,
$$F(\theta x,\theta y) = \theta^r F(x,y)$$
where would we refer to $F$ being a homogeneous function of degree $r$. Here it is much clearer to see that if $r>1$, we have increasing returns to scale because for a given $\...