Quaternion kinematics provides a singularity-free and computationally efficient model for modeling rotational motion in aerospace, robotics, navigation, and computer vision systems. The continuous-time quaternion kinematics model provides an accurate representation of the state changes of a rigid body under angular velocity inputs. However, practical implementations in digital control systems, estimation algorithms, and embedded platforms and systems require linear or discrete approximations of the original nonlinear continuous-time model. These approximations inevitably introduce modeling errors, numerical drift, and stability concerns that may degrade system performance. In this paper, various linear and discrete models of quaternion kinematics are evaluated. In this regard, first, the relationships between the exact continuous-time quaternion differential equation and its linear and discrete representations, which have been introduced and used in various references, are derived. Then, the approximation errors, stability properties, and normalization limits are analyzed under varying angular velocity values and different sampling intervals. This analysis highlights the relationship between computational efficiency and accuracy, especially in highly dynamic rotational regimes. Finally, the limitations of simplified linear and discrete models are discussed in the context of their application in state estimation filters, controller design, and real-time embedded systems. As a result, quantitative and structural insights are provided on when linear and discrete quaternion models remain valid and when higher-order or more precise integration schemes are necessary.