"Kinetics" is just a synonym for "dynamics," and most English books use the term "dynamics" rather than "kinetics." There are plenty of mechanical engineering textbooks with "dynamics" in the title, but few with "kinetics" (I'm ignoring books on topics like "kinetics of chemical reactions", etc, which are not about mechanical engineering)
The difference between "kinetics" and "kinematics" is that kinematics deals with the motion of object without considering what causes that motion (for example the mathematical relationships between position, velocity, and acceleration of a particle moving along a given curved path in space), but kinetics/dynamics does consider the forces causing the motion (i.e. Newton's laws of motion).
Since a fundamental topic in studying mechanical vibrations is the fact that for linear elastic materials which obey Hooke's law, Newton's laws mean that the motion for small displacements about an equilibrium position is simple harmonic motion, I would classify most of the subject of "mechanical vibrations" to be part of dynamics/kinematics, rather than kinetics.