Would the integral calculated by the PID controller be considered definite or indefinite?
According to the definition given in this WP article, the relevant integral is of the form
If this is regarded as a function of $t$ (say in a theoretical treatment), it would be an indefinite integral; however, your question concerns the integral calculated by the PID controller, and this is a definite integral for each specified value of $t$.
Depends on the control circuit but hardly ever a precise integration.
If the controller is realized as a RLC circuit (an analog controller) then maybe you might say it is a definite integral but they most often come with a reset (anti-windup) algorithm to avoid saturation and kickback hence the integral is taken from the reset time to the actual time.
On the other hand if it is a discrete time controller, you first filter it out so $e(t)$ gets convoluted with a low-pass filter and then due to saturation problems you might wish to make the $I$ parameter gain dependent. If the error is too large you decrease the $I$ parameter of the controller etc.
Therefore, theoretically it would be a definite integral but to avoid oscillations around the reference signal and for other practical matters it is almost never implemented as a pure integrator.