Radio-loud quasars (RLQs) are more X-ray luminous than predicted by the X-ray-optical/UV relation for radio-quiet quasars (RQQs). The excess X-ray emission depends on the radio-loudness parameter (R) and radio spectral slope. We construct a uniform sample of 729 optically selected RLQs with high fractions of X-ray detections and measurements. We find that steep- spectrum radio quasars (SSRQs) follow a quantitatively similar Lx-Luv relation as that for RQQs, suggesting a common coronal origin for the X-ray emission of both SSRQs and RQQs. However, the corresponding intercept of SSRQs is larger than that for RQQs and increases with R, suggesting a corona-jet connection. Flat-spectrum radio quasars (FSRQ) are generally more X-ray luminous than SSRQs at given Luv and R. A distinct jet component is likely important for only a small portion of FSRQs. The corona-jet, disk-corona, and disk-jet connections of RLQs are likely driven by independent physical processes. Furthermore, the corona-jet connection implies that small-scale processes in the vicinity of SMBHs (probably associated with the magnetic flux instead of black-hole spin) are controlling the radio-loudness of quasars.