Yes, in the context of two inertial frames of reference, there is a clear and obvious element that determines which frame is moving and which is stationary: relative motion. The concept of relative motion refers to the motion of an object or observer in relation to another object or observer.
When comparing two inertial frames of reference, we can determine their relative motion by examining the motion of objects or observers in each frame. If an object or observer appears to be at rest or moving uniformly in a straight line within a frame, while objects or observers in the other frame appear to be in motion relative to it, then the frame with the stationary or uniformly moving object/observer is considered the stationary frame, and the other frame is considered the moving frame.
It's important to note that the concept of motion is always relative, and there is no absolute reference frame that can be identified as truly stationary. Motion can only be defined in relation to other objects or observers. Therefore, when discussing motion between two inertial frames, the determination of which frame is moving and which is stationary is based on their relative motion.