We investigate the difference between those relativistic models based on
interpreting a realistic nucleon-nucleon interaction as a perturbation of the
square of a relativistic mass operator and those models that use the method of
Kamada and Gl\"ockle to construct an equivalent interaction to add to the
relativistic mass operator. Although both models reproduce the phase shifts and
binding energy of the corresponding non-relativistic model, they are not
scattering equivalent. The example of elastic electron-deuteron scattering in
the one-photon-exchange approximation is used to study the sensitivity of
three-body observables to these choices. Our conclusion is that the differences
in the predictions of the two models can be understood in terms of the
different ways in which the relativistic and non-relativistic S-matrices are
related. We argue that the mass squared method is consistent with conventional
procedures used to fit the Lorentz-invariant cross section as a function of the
laboratory energy.Comment: Revtex 13 pages, 5 figures, corrected some typo
