understanding time constant meaning in signal processing
There is no general formula to calculate time constants for all signals. Time constants are defined for some signals. For your exponential signal

\begin{equation}
e^{-a \cdot t} = e^{-t / c}
\end{equation}

it just means that always after the time $c$ has passed, the value of the function has been reduced by $1 / e$. A more common interpretation is half-life defined as

$$t_{1/2} = \frac{ln(2)}{a}$$

which is the time after the function has gone from 1 (at time zero) to 1/2 (at time $t_{1/2}$) and so on.

However, there is no general time constant that can be applied to any signal, because, as you already mentioned, it is unclear what it represents.