  We show how to properly extract the local charge order in two-dimensional
materials from scanning tunneling microscopy/spectroscopy (STM/STS)
measurements. When the charge order presents spatial variations at the atomic
scale inside the unit cell and is energy dependent, particular care should be
taken. In such cases the use of the lock-in technique, while acquiring an STM
topography in closed feedback loop, leads to systematically incorrect dI/dV
measurements giving a false local charge order. A correct method is either to
perform a constant height measurement or to perform a full grid of dI/dV(V)
spectroscopies, using a bias voltage setpoint outside the material bandwidth
where the local density-of-states (LDOS) is spatially homogeneous. We take as a
paradigmatic example of two-dimensional material the 1/3 single-layer
Pb/Si(111). As large areas of this phase cannot be grown, charge ordering in
this system is not accessible to angular resolved photoemission or grazing
x-ray diffraction. Previous investigations by STM/STS supplemented by {\it ab
initio} Density Functional Theory (DFT) calculations concluded that this
material undergoes a phase transition to a low-temperature $3\times 3$
reconstruction where one Pb atom moves up, the two remaining Pb atoms shifting
down. A third STM/STS study by Adler {\it et al.} [PRL 123, 086401 (2019)] came
to the opposite conclusion, i.e. that two Pb atoms move up, while one Pb atom
shifts down. This latter erroneous conclusion comes from a misuse of the
lock-in technique. In contrast, using a full grid of dI/dV(V) spectroscopy
measurements, we show that the energy-dependent LDOS maps agree very well with
state-of-the-art DFT calculations confirming the one-up two-down charge
ordering. This structural and charge re-ordering in the $3\times 3$ unit cell
is equally driven by electron-electron interactions and the coupling to the
substrate.
