In this paper, single-photon avalanche diode (SPAD) photodetectors are leveraged across long-range underwater optical wireless communication (UOWC) systems for non-turbid waters, i.e., pure sea and clear ocean, in weak turbulent conditions. We derive the bit error probability of the system using on-off keying (OOK) along with two types of SPADs: ideal, i.e., with zero dead time, and practical, i.e., with non-zero dead time. In our investigations for OOK systems, we study the effect of using both the optimum threshold (OTH) and constant threshold (CTH) at the receiver. Furthermore, we analyze the performance of systems using binary pulse position modulation (B-PPM) and compare them with the ones using OOK. Our results are presented for practical SPADs and both active and passive quenching circuits. We demonstrate that OOK systems with OTH perform slightly better than B-PPM. However, our investigations reveal that in turbulent conditions where using OTH may be challenging, employing B-PPM can be preferable to OOK.