  The non-holonomic deformation of the nonlinear Schr\"odinger equation,
uniquely obtained from both the Lax pair and Kupershmidt's bi-Hamiltonian
[Phys. Lett. A 372, 2634 (2008)] approaches, is compared with the
quasi-integrable deformation of the same system [Ferreira et. al. JHEP 2012,
103 (2012)]. It is found that these two deformations can locally coincide only
when the phase of the corresponding solution is discontinuous in space,
following a definite phase-modulus coupling of the non-holonomic inhomogeneity
function. These two deformations are further found to be not gauge-equivalent
in general, following the Lax formalism of the nonlinear Schr\"odinger
equation. However, asymptotically they converge for localized solutions as
expected. Similar conditional correspondence of nonholonomic deformation with a
non-integrable deformation, namely, due to local scaling of the amplitude of
the nonlinear Schr\"odinger equation is further obtained.
