In this study, linear transceiver designs are proposed based on modified definitions of signal-to-leakage-plus-noise ratio (mSLNR) for multi-cell coordinated beamforming scenarios. Two mSLNR definitions are presented, one of which modifies the computation of leakage and noise powers, while the other applies the modified procedure to the desired power calculation in addition to the computation of leakage and noise powers. The proposed mSLNRs are generally valid for any receiver types. Various linear receivers are studied, several of which, when applied with the first mSLNR definition, are shown to be equivalent. Moreover, a special case of the proposed scheme is shown to be equivalent to an existing algorithm. Iterative precoding algorithms are also presented with a discussion on their convergence properties. The proposed transceiver designs are further extended to incorporate scheduling functionalities. A weighted sum rate maximisation problem is formulated and is solved by two separate approaches, namely power allocation and data-substream selection methods. Simulation results show that, for multiple users per cell, the proposed algorithms can effectively integrate user and substream selections and achieve multi-user diversity gain.