The sign of the Kerr nonlinear coefficient has long been regarded as irrelevant to the direction of the Raman-induced soliton self-frequency shift. Yet the standard generalized nonlinear Schr¨odinger equation (GNLSE) predicts a frequency shift that depends on the sign of the nonlinearity, which leads to an unphysical blue shift in the defocusing case. We resolve this inconsistency by deriving the time-domain photon-conserving GNLSE (pcGNLSE) from its established frequency-domain counterpart. The derivation identifies additional carrier-frequency terms in the Kerr and Raman operators. The resulting energy balance predicts a decrease in localized optical pulse energy, independent of the sign of the Kerr coefficient or its frequency slope. Applying the method of moments, we derive five-parameter evolution equations for bright and dark solitons and obtain constant-peak-power and constant-blackness conditions that identify Raman-soliton attractors in both focusing and defocusing Kerr media. These analytical conditions relate pulse parameters directly to dispersion, nonlinear dispersion, and Raman corrections, providing a physically consistent framework for designing Ramansoliton dynamics in emerging semiconductor waveguide and microresonator platforms.