convolution integral limits

Suppose that $f$ and $g$ are defined such that they only can take non-zero values on the positive reals, but are zero everywhere on the negative reals. Then

$$\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\; d\tau = \int_0^t f(\tau)g(t-\tau)\; d\tau$$

I think the second case you came across is just a manifestation of your functions being defined that way. That's often implicit in the theory of Laplace transforms for instance or in probability theory (think of eg. the exponential distribution).