This paper provides precise performance analysis of the dual-hop mixed radio frequency (RF)/unified free space optical (FSO) decode-and-forward (DF) relaying system, in which the heterodyne detection and the intensity modulation-direct detection (IM-DD) are taken into account for FSO detection. To this end, we derive closed-form expressions for the outage probability, average bit error rate (BER), and the ergodic channel capacity of this system. In this analysis, we utilize, for the first time as per our knowledge, a precise channel capacity result for the IM-DD channel. Moreover, this is the first time that not only the (IM-DD input-independent) but also the (IM-DD cost-dependent) additive white Gaussian noise (AWGN) channel is considered in such system analyses. Additionally in this study, we assume that the first hop (RF link) follows Nakagami-m fading, and the second hop (FSO link) follows Málaga (M) turbulence with pointing errors. These fading and turbulence models contain other models (such as Rayleigh and Gamma-Gamma) as special cases, thus, our analyses can be seen as a generalized one from the RF and FSO fading models point of view. Also, in BER derivation, we assume that the modulation schemes in the two hops can be different, since not all modulation schemes are suitable for IM-DD FSO links. In addition, the system performance is investigated asymptotically at high signal to noise ratios. This investigation leads to new nonreported coding gain and diversity order analyses of such system. Interestingly, we find that in the FSO hop, at high transmitted powers, all the considered FSO detectors result in the same diversity order. Furthermore, we provide simulation results that verify the accuracy of the obtained analytical and asymptotic expressions.