Human operation errors often lead to collisions involving mobile vehicles. Zeroing Control Barrier Functions (ZCBFs) are widely studied as a tool for guaranteeing safety. However, their definitions and conditions lack a unified framework. In this study, we propose a Relaxed ZCBF (R-ZCBF), which unifies and extends existing ZCBF theories by relaxing their conditions. This approach enables safer and more flexible control, expanding ZCBF applicability to a broader range of systems.