  The overlap, or similarity, between liquid configurations is at the core of
the mean-field description of the glass transition, and remains a useful
concept when studying three-dimensional glass-forming liquids. In liquids,
however, the overlap involves a tolerance, typically of a fraction $a/\sigma$
of the inter-particle distance, associated with how precisely similar two
configurations must be for belonging to the same physically relevant "state".
Here, we systematically investigate the dependence of the overlap fluctuations
and of the resulting phase diagram when the tolerance is varied over a large
range. We show that while the location of the dynamical and thermodynamic glass
transitions (if present) is independent of $a/\sigma$, that of the critical
point associated with a transition between a low- and a high-overlap phases in
the presence of an applied source nontrivially depends on the value of
$a/\sigma$. We rationalize our findings by using liquid-state theory and the
hypernetted chain (HNC) approximation for correlation functions. In addition,
we confirm the theoretical trends by studying a three-dimensional glass-former
by computer simulations. We show in particular that \rev{a range of $a/\sigma$
below what is commonly considered maximizes the temperature of the critical
point, pushing it up in a liquid region where viscosity is low and computer
investigations are easier due to a significantly faster equilibration.
