If the derivative is calculated as change in function with respect to change in the independent variable then it should be $\frac{dq}{dt} = \lim_{\Delta t \to 0} \frac{\Delta q}{\Delta t}$ but why is it defined as $\frac{dq}{dt} = \lim_{\Delta t\to0} \frac{q_{t + \Delta t} - q_{t}}{\Delta t}$ ?
It appears you are being confused by a conflict of notation.
By usual conventions, if we have two variables $t$ and $\Delta t$ and $q$ is a function of one variable, we define the notation $\Delta q(t)$ to mean $q(t + \Delta t) - q(t)$.
Similarly, if we have a variable $y$ related to $t$ by a function $y=f(t)$, then we define a new variable $\Delta y$ that satisfies $\Delta y = f(t + \Delta t) - f(t)$.
It is this convention that is being invoked when we use shorthand for difference quotients; e.g. as in equations like
$$ f'(t) = \lim_{\Delta t \to 0} \frac{\Delta f(t)}{\Delta t} \qquad \qquad \qquad \qquad \frac{\mathrm{d}y}{\mathrm{d}t} = \lim_{\Delta t \to 0} \frac{\Delta y}{\Delta t} $$
The variable $\Delta q$ appearing in this theorem, however, has nothing to do with that convention.