  Orbital eccentricity in compact binaries is considered to be a key tracer of
their astrophysical origin, and can be inferred from gravitational-wave
observations due to its imprint on the emitted signal. For a robust
measurement, accurate waveform models are needed. However, ambiguities in the
definition of eccentricity can obfuscate the physical meaning and result in
seemingly discrepant measurements. In this work we present a suite of 28 new
numerical relativity simulations of eccentric, aligned-spin binary black holes
with mass ratios between 1 and 6 and initial post-Newtonian eccentricities
between 0.05 and 0.3. We then develop a robust pipeline for measuring the
eccentricity evolution as a function of frequency from gravitational-wave
observables that is applicable even to signals that span at least $\gtrsim 7$
orbits. We assess the reliability of our procedure and quantify its robustness
under different assumptions on the data. Using the eccentricity measured at the
first apastron, we initialise effective-one-body waveforms and quantify how the
precision in the eccentricity measurement, and therefore the choice of the
initial conditions, impacts the agreement with the numerical data. We find that
even small deviations in the initial eccentricity can lead to non-negligible
differences in the phase and amplitude of the waveforms. However, we
demonstrate that we can reliably map the eccentricities between the simulation
data and analytic models, which is crucial for robustly building eccentric
hybrid waveforms, and to improve the accuracy of eccentric waveform models in
the strong-field regime.
