In this work, a generalized approach is proposed to review the performance of relaying designs with energy harvesting capability. The unified modeling of generalized energy harvesting relaying (GEHR) design covers the non-energy harvesting designs and the well-known energy harvesting designs, i.e., time-based relaying (TR) and power-based relaying (PR). Moreover, the hybrid of both TR and PR designs is also catered. We find the mathematical representations for the outage probability, ergodic capacity and average throughput in Rayleigh fading channels for amplify-and-forward (AF) and decode-and-forward (DF) relaying modes. The closed-form expressions are derived for the outage probability. To validate that GEHR design is a generalization of TR and PR designs, we study the individual cases of GEHR design from the perspectives of schematic diagram, signal analysis and the performance evaluation parameters comparison. Furthermore, the GEHR design is studied for the mixed Rayleigh-Rician fading channels. We considered two sub-cases of mixed fading, i.e., in case 01, source to relay (SR) link is considered as Rayleigh channel and relay to destination (RD) link is considered as a Rician channel. Conversely, in case 02, SR link is taken as Rician channel and RD link is a Rayleigh channel. We find the mathematical expressions for the ergodic capacity, outage probability and average throughput for DF and AF relaying for both cases of mixed fading channels. The analytical results in both channel configurations are presented for throughput and verified using extensive Monte-Carlo simulations. The results show that the proposed GEHR design can be set to work as not only for the conventional TR and PR designs but also for hybrid of them. Furthermore, with some slight modifications in the proposed design, it can work as a conventional non-energy harvesting cooperative relaying model.