This paper revisits the quantum-amplitude-based derivation of the gravitational waveform from the scattering of two spinless massive bodies at the third order in Newton's constant, $ h \sim G + G^2 + G^3 $, and compares it with the classical multipolar-post-Minkowskian (MPM) counterpart. The authors find complete agreement between the two results up to the fifth order in the small velocity expansion after accounting for three key aspects of the amplitude derivation. These include a frame rotation by half the classical scattering angle, an additional finite term from dimensional regularization of infrared divergences, and contributions from zero-frequency gravitons in the observable-based formalism. The paper discusses the MPM formalism, which computes the time-domain waveform as a sum of irreducible multipolar contributions. The MPM formalism is set up in the center-of-mass (c.m.) frame and uses a retarded time variable to describe the waveform. The waveform is expressed in terms of multipole moments, which are computed using the stress-energy tensor of the source. The MPM formalism includes both linear and nonlinear contributions to the waveform, with the nonlinear terms arising from couplings between multipole moments. The authors also discuss the observable-based formalism, which constructs quantum scattering observables by comparing the expectation value of a hermitian operator between the initial and final states of the scattering process. The observable-based formalism is used to compute the scattering waveform by considering the linearized Riemann tensor or the transverse-traceless components of the asymptotic metric fluctuations. The results from the observable-based formalism are compared with those from the MPM formalism, and the authors find complete agreement up to the available post-Newtonian accuracy. The paper also discusses the role of zero-frequency gravitons in the observable-based formalism, which contribute additional terms at both $ h \sim G $ and $ h \sim G^3 $ when including disconnected diagrams. These contributions are interpreted as a Bondi-Metzner-Sachs (BMS) supertranslation of the result with no contributions from such states. The authors also discuss the importance of a subtle $ O(\epsilon/\epsilon) $ contribution from the graviton being described by a symmetric, traceless $ (d-2)\times(d-2) $ matrix in $ d $ dimensions. The paper concludes with a detailed comparison of the MPM and observable-based formalisms, showing that they agree up to the available post-Newtonian accuracy. The authors also discuss the importance of the frame rotation and the role of zero-frequency gravitons in the observable-based formalism. The results demonstrate the consistency between the quantum-amplitude-based and classical multipolar-post-Minkowskian approaches to gravitational waveforms.This paper revisits the quantum-amplitude-based derivation of the gravitational waveform from the scattering of two spinless massive bodies at the third order in Newton's constant, $ h \sim G + G^2 + G^3 $, and compares it with the classical multipolar-post-Minkowskian (MPM) counterpart. The authors find complete agreement between the two results up to the fifth order in the small velocity expansion after accounting for three key aspects of the amplitude derivation. These include a frame rotation by half the classical scattering angle, an additional finite term from dimensional regularization of infrared divergences, and contributions from zero-frequency gravitons in the observable-based formalism. The paper discusses the MPM formalism, which computes the time-domain waveform as a sum of irreducible multipolar contributions. The MPM formalism is set up in the center-of-mass (c.m.) frame and uses a retarded time variable to describe the waveform. The waveform is expressed in terms of multipole moments, which are computed using the stress-energy tensor of the source. The MPM formalism includes both linear and nonlinear contributions to the waveform, with the nonlinear terms arising from couplings between multipole moments. The authors also discuss the observable-based formalism, which constructs quantum scattering observables by comparing the expectation value of a hermitian operator between the initial and final states of the scattering process. The observable-based formalism is used to compute the scattering waveform by considering the linearized Riemann tensor or the transverse-traceless components of the asymptotic metric fluctuations. The results from the observable-based formalism are compared with those from the MPM formalism, and the authors find complete agreement up to the available post-Newtonian accuracy. The paper also discusses the role of zero-frequency gravitons in the observable-based formalism, which contribute additional terms at both $ h \sim G $ and $ h \sim G^3 $ when including disconnected diagrams. These contributions are interpreted as a Bondi-Metzner-Sachs (BMS) supertranslation of the result with no contributions from such states. The authors also discuss the importance of a subtle $ O(\epsilon/\epsilon) $ contribution from the graviton being described by a symmetric, traceless $ (d-2)\times(d-2) $ matrix in $ d $ dimensions. The paper concludes with a detailed comparison of the MPM and observable-based formalisms, showing that they agree up to the available post-Newtonian accuracy. The authors also discuss the importance of the frame rotation and the role of zero-frequency gravitons in the observable-based formalism. The results demonstrate the consistency between the quantum-amplitude-based and classical multipolar-post-Minkowskian approaches to gravitational waveforms.