I think I have solved the problem after thinking about it.
Basically, I forgot the mathematical formalism for a moment and thought about it physically. We want to know the general velocity and position of a body that is moving at constant acceleration. Therefore, all we need is one moment in time for which we know the position and velocity, then we can use the constant acceleration to get other times. In other words, we know $v(t_0)$, so we just do $v(t_0)+at$ for times more than $t_0$, and $v(t_0)-at$ for times less than $t_0$. This is without using any "mathematics", since we know that each second, the velocity increases by $a$ since it is constant acceleration.
Now let's see how this compares with the integral formalism. As it turns out, $t_0$ is not an initial time at all, it is only an arbitrary time, as it clear from the intuitive physical explanation above. It is only when we come to the mathematics that it all of a sudden becomes an "initial" time. Do you know why? Because in the mathematical formalism, when we work out the area under the a-t graph, we have to choose an "initial" time on the graph from which we work out the area. 499
It couldn't be otherwise, since area is 2-D and we have to start somewhere, ie $\int_{t_0}^t$. If there was some magical mathematical way we could avoid starting somewhere, and invent a notation where we don't appear to "start" at $t_0$, say $\zeta(t_0)$ (this isn't like $\int_{t_0}^t$ which has the illusion of "starting" at $t_0$ and moving to $t$), then it would be as clear as the physical explanation above. (However, I guess we can rest contented with the fact that $\int_{t_0}^t = -\int _t^{t_0}$, so that it does have symmetry after all, and does not have to have an illusion of starting at $t_0$). Unfortunately, this is not the case. So the mathematical formalism makes it look like $t_0$ is an "initial" time. It is not: it is simply an arbitrary time for which we know v and x.
So it isn't as simple as I thought at first. The fact that authors call it "initial" and "starting" time is due to the mathematics, in particular, how it appears on the graph. It has nothing to do with the physics. 499