  Galaxy-scale gravitational lenses are often modeled with two-component mass
profiles where one component represents the stellar mass and the second is an
NFW profile representing the dark matter. Outside of the spherical case, the
NFW profile is costly to implement, and so it is approximated via two different
methods; ellipticity can be introduced via the lensing potential (NFWp) or via
the mass by approximating the NFW profile as a sum of analytical profiles
(NFWm). While the NFWp method has been the default for lensing applications, it
gives a different prescription of the azimuthal structure, which we show
introduces ubiquitous gradients in ellipticity and boxiness in the mass
distribution rather than having a constant elliptical shape. Because unmodeled
azimuthal structure has been shown to be able to bias lens model results, we
explore the degree to which this introduced azimuthal structure can affect the
model accuracy. We construct input profiles using composite models using both
the NFWp and NFWm methods and fit these mocks with a power-law elliptical mass
distribution (PEMD) model with external shear. As a measure of the accuracy of
the recovered lensing potential, we calculate the value of the Hubble parameter
$H_0$ one would determine from the lensing fit. We find that the fits to the
NFWp input return $H_0$ values which are systematically biased by about $3\%$
lower than the NFWm counterparts. We explore whether such an effect is
attributable to the mass sheet transformation (MST) by using an MST-independent
quantity, $\xi_2$. We show that, as expected, the NFWm mocks are degenerate
with PEMD through an MST. For the NFWp, an additional bias is found beyond the
MST due to azimuthal structures {\it exterior to the Einstein radius}. We
recommend modelers use an NFWm prescription in the future, such that azimuthal
structure can be introduced explicitly rather than implicitly.
