When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model is balanced truncation (BT), a method which relies on simultaneously diagonalizing the system Gramians and which has been extensively studied for deterministic linear systems. This balancing procedure has already been extended to bilinear equations [S. A. Al-Baiyat and M. Bettayeb, Proceedings of the 32nd IEEE Conference on Decision and Control, 1993, pp. 22-27], an important subclass of nonlinear systems. We call the approach of Al-Baiyat and Bettayeb type I balancing. We provide an alternative generalization of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighborhood of zero. Furthermore, we provide an H-infinity-error bound which so far is not known when applying type I BT to bilinear systems.