  Orbital eccentricity is a key signature of dynamical binary black hole
formation. The gravitational waves from a coalescing binary contain information
about its orbital eccentricity, which may be measured if the binary retains
sufficient eccentricity near merger. Dedicated waveforms are required to
measure eccentricity. Several models have been put forward, and show good
agreement with numerical relativity at the level of a few percent or better.
However, there are multiple ways to define eccentricity for inspiralling
systems, and different models internally use different definitions of
eccentricity, making it difficult to directly compare eccentricity
measurements. In this work, we systematically compare two eccentric waveform
models, $\texttt{SEOBNRE}$ and $\texttt{TEOBResumS}$, by developing a framework
to translate between different definitions of eccentricity. This mapping is
constructed by minimizing the relative mismatch between the two models over
eccentricity and reference frequency, before evolving the eccentricity of one
model to the same reference frequency as the other model. We show that for a
given value of eccentricity passed to $\texttt{SEOBNRE}$, one must input a
$20$-$50\%$ smaller value of eccentricity to $\texttt{TEOBResumS}$ in order to
obtain a waveform with the same empirical eccentricity. We verify this mapping
by repeating our analysis for eccentric numerical relativity simulations,
demonstrating that $\texttt{TEOBResumS}$ reports a correspondingly smaller
value of eccentricity than $\texttt{SEOBNRE}$.
