If you define a wheel that can only rotate and move in the direction it is pointing to but using a combination of motions (arbitrarily short) it can be moved sideways, does the system still remain holonomic, by definition?
In robotics, a holonomic chassis is one that can move in an arbitrary direction regardless of the robot's facing. Such a robot can move in interesting ways. For example, if you have a fixed camera on a wheeled holonomic robot, the robot could smoothly move, turn, park, etc. without turning the camera.
So if the wheel has to change direction before moving in a given direction, then the robot is nonholomic.
The only ones I've seen use omni-wheels or Mecanum wheels. Different programming is needed for both types.
If you can know the position/orientation of the wheel by measuring the total angular rotation, it would be holonomic. Since the order in which the “small” rotations matters in your example, you do not know the pose of the wheel just based on total rotation. Therefore it is not a geometric relationship between input actuations and output pose; it is non-holonomic. At best you can define a differential relationship.