Reconfigurable intelligent surface (RIS)-assisted power-domain non-orthogonal multiple access (PD-NOMA) system has emerged as a revolutionary technology to enhance the spectrum efficiency for future wireless networks. This work introduces two novel RIS systems, namely, RIS partition-assisted (RISP) PD-NOMA (RISP-PD-NOMA) and RISP-Quadrature NOMA (RISP-Q-NOMA), to improve the signal quality of all users by dedicating fixed RIS units to each user for phase cancellation. The closed-form expressions of average sum-rate, outage probability, and diversity order of both systems are evaluated under the Rician fading channel for perfect and imperfect successive interference cancellation (SIC). Further, the performance of both systems is compared with RIS-division PDNOMA (RISD-PD-NOMA) system where the RIS subsurface assigned to one user is not exposed to another user. It is noticed from the analysis that under perfect SIC, RISP-PD-NOMA outperforms RISP-Q-NOMA and RISD-PD-NOMA. However, under imperfect SIC, RISP-Q-NOMA demonstrates superior performance than other systems, due to lesser number of SIC operations that lead to less stringent constraints on the power allocation, reduce detection delay, and improve SIC stability. Furthermore, the analytical expression for bit error rate (BER) of RISP-PD-NOMA and RISP-Q-NOMA is derived, and then closed-form expression of average-BER is evaluated. Numerical results demonstrate that RISP-Q-NOMA is superior to RISPPD-NOMA and RISD-PD-NOMA.