  The GRB central engine intrinsic activity time $T_{\rm ce}$ is usually
described through either the $\gamma$-ray duration $T_{90}$ or through a
generalized burst duration $t_{\rm burst}$ which includes both the $\gamma$-ray
emission and (when present) an extended flaring X-ray plateau. Here, we define
a more specific operational description of $T_{\rm ce}$, and within the
framework of the internal-external shock model, we develop a numerical code to
study the relationship between $T_{90}$ and $T_{\rm ce}$, as well as between
$t_{\rm burst}$ and $T_{\rm ce}$, for different initial conditions. We find
that when $T_{\rm ce}\lesssim 10^4$ s, late internal collisions or refreshed
external collisions result in values of $T_{\rm 90}$ and $t_{\rm burst}$ larger
than $T_{\rm ce}$, usually by factors of $2-3$. For $T_{\rm ce}\gtrsim 10^4$ s,
the $t_{\rm burst}$ is always a good estimator for $T_{\rm ce}$, while $T_{90}$
can underpredict $T_{\rm ce}$ when the late central engine activity is
moderate. We find a clear bimodal distribution for $T_{\rm ce}$, based on our
simulations as well as on the observational data for $T_{90}$ and $t_{\rm
burst}$. We suggest that $t_{\rm burst}$ is a reliable measure for defining
"ultra-long" GRBs. Bursts with $T_{90}$ of order $10^3$ s need not belong to a
special population, while bursts with $t_{\rm burst} > 10^4$ s, where the late
central engine activity is more moderate and shows up in X-rays, may represent
a new population. These conclusions are insensitive to the initial conditions
assumed in the models.
