Radio-loud quasars (RLQs) are more X-ray luminous than predicted by the X-ray-optical/UV relation (i.e. L_x_{prop.to}L_uv_^{gamma}^) for radio-quiet quasars (RQQs). The excess X-ray emission depends on the radio-loudness parameter (R) and radio spectral slope ({alpha}_r_). We construct a uniform sample of 729 optically selected RLQs with high fractions of X-ray detections and {alpha}_r_ measurements. We find that steep-spectrum radio quasars (SSRQs; {alpha}_r_=<-0.5) follow a quantitatively similar L_x_{prop.to}L_uv_^{gamma}^ 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 connection between the radio jets and the configuration of the accretion flow. Flat-spectrum radio quasars (FSRQs; {alpha}_r_>-0.5) are generally more X-ray luminous than SSRQs at given L_uv_ and R, likely involving more physical processes. The emergent picture is different from that commonly assumed where the excess X-ray emission of RLQs is attributed to the jets. We thus perform model selection to compare critically these different interpretations, which prefers the coronal scenario with a corona-jet connection. A distinct jet component is likely important for only a small portion of FSRQs. The corona-jet, disc-corona, and disc-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 supermassive black holes, probably associated with the magnetic flux/topology instead of black hole spin, are controlling the radio-loudness of quasars.