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Planar motion:A rigid body B is said to be in planar motion if each particle of B moves in a fixed plane and all these planes are parallel to each other.
This is my understanding of this definition. If a motion of rigid body has 2D motion then It is said to be planar. I think including the idea of plane is unnecessary but I am sure I kinda didn't get the definition properly. So please help!
Your intuitive understanding seems reasonable, but it is not rigorous enough to be considered a definition. After all, it would be strange if you could define planar motion without resorting to the concept of the plane.
Indeed, when you define planar motion in terms of "2D motion", you failed to define what "2D motion" is. Can we allow it to be a motion constrained on any 2D surface? If so, this would include motion on a sphere or a torus, which is definitely not planar motion. So it must be more restricted than this. In this context you can either say: "2D motion is planar motion", which is circular reasoning, or "2D motion is motion constrained on a plane", which is where the concept of the plane was hiding all along.
In the "Arya" and "Simon" classical mechanics,we explain the rigid body as some particles often in case of Avogadro number and distance of between each 2 particles is fixed and based on Newton third law of motion,force between particles will be zero.we do not have any ideal rigid body in nature that always above conditions are true but most of the time, with some Approximation,we say that they are rigid body.another point about rigid bodies is that they have a point called "center of mass" that we can imagine all the mass of the object is in the center of mass and we can re-write the laws of the motion for the point instead of whole rigid body. hope to be useful but it is better to take a look to the classical mechanics references.
A rigid body B is said to be in planar motion if each particle of B moves in a fixed plane and all these planes are parallel to each other.
Where did you pick up this definition? In my opinion, planar motion of a rigid body means that only the center of gravity of the body is moving in a fixed plane but not necessarily all of it's particles.
According to the definition you posted, a rotating body cannot be in planar motion (except the rotation is around an axis perpendicular to the plane of motion).
