The system you're referring to is called the "Conditional Turn-Based Battle" (CTB) system, but more specifically, the feature that allows for special actions or attacks based on certain conditions is known as a "Context-Dependent Battle System".

This system was popularized by Final Fantasy X, but other games have implemented similar systems, such as the "Active Time Battle" (ATB) system with conditional actions in games like Final Fantasy VII and IX, or the "Tactical Turn-Based" system with context-dependent actions in games like Tactics Ogre and Vagrant Story.

The CTB system has been influential in the development of turn-based battle systems, and its elements can be seen in many modern RPGs.