With progress in cooperative and autonomous driving, there is an increasing interest in intelligent intersections to replace conventional traffic lights and, thereby, improve traffic efficiency. To avoid accidents in such safety-critical systems, a traffic protocol needs to be implemented. In this article, we are concerned with synchronous traffic protocols, i.e., those that synchronize the arrival time of vehicles at the intersection. In particular, such protocols are normally conceived for homogeneous vehicles of approximately the same size/length. However, these do not extend well to heterogeneous vehicles, i.e., they lead to unviable requirements on the road infrastructure. To overcome this limitation, based on the observation that large/overlength vehicles like buses and trams are less frequent than passenger vehicles, we propose an approach that treats them as exceptions (rather than the rule) leading to a much more efficient design. In contrast to approaches from the literature, we implement a two-speed policy—with a high speed for drive-through and a low speed for turn maneuvers—and analyze both single-vehicle as well as fairness-based platoon crossing. To conclude, we perform detailed comparisons illustrating the benefits by the proposed approach.