Suppose you have body with mass M experiencing a force F. If you model the force as being applied to the whole body, then this gives an acceleration F/M. Now suppose you cut this body into n equal parts. The part that the force is directly applied to will experience force F over mass M/n. So if you model the force as being applied just to one part, that appears to give an acceleration of nF/M. Contradiction? No. We're modeling this as a rigid body. So if one part accelerates, it must take the rest of the body with it. The rest of the body has mass M(n-1)/n. If this mass is accelerating at F/M, then it is experiencing a force of (F/M)(M(n-1)/n) = F(n-1)/n. So if we're modeling the force as applying only to one part, the rest of the body isn't being accelerated by this force, at least not directly. Instead, it must be being accelerated by the part that has the force applied to it. So that part is applying a force of F(n-1)/n to the rest of the body, and by Newton's Third Law, the rest of the body is applying a force of F(n-1)/n to that part. So the total resultant force for the part with an external force is the external force F minus the force F(n-1)/n from the rest of the body, which gives F/n. A force of F/n applied to a mass of M/n gives an acceleration of F/M.
If you want, you can break this analysis down even further. Suppose you have a object of n sections, and you apply a force to a section. Then that section will experience the exterior force plus forces from the adjacent sections. Those sections will experience a force from that original section, plus forces their adjacent sections. And so on. You can then set up a system of equations involving all these forces, relate the acceleration of each part to the net force it experiences, apply the constraint that the acceleration on each part is equal to the acceleration of each other part, and solve that system of equations. Intuitively, we should find that the acceleration for each part is F/M, regardless of how we divide the body into "parts".
Or we can simply recognize that we are dealing with a model to begin with, and different aspects of the model apply to different situations. We can treat the body as being made up of differential masses for such purposes as calculating moments of inertia, but a single mass for calculating acceleration from force. We can treat the body as being a point mass for purposes of calculating what gravitational force it creates, and yet treat forces as being displaced from the center of mass for purposes of calculating torque.
This would mean that the definition of a force on a rigid body is different to that of a force on a system of particles (since if it is acting on the rigid body as a whole it is not acting at a single point)
The definition of force is the same. What model we use depends on the context. The term "rigid body" does not refer to any actual physical object. Forces are transmitted through an object at the speed of sound in that object, which cannot be higher than c. No object is truly "rigid". When we talk about a "rigid object", that is simply shorthand for "an object that, in the scenario currently under consideration, deviates so little from the theoretical ideal of a rigid body that it can be modeled as a rigid body to the degree of accuracy required". This is similar to how we talk about applying the ideal gas law to gases, even though there is no such thing as an ideal gas in the real world. So when we model something as a rigid body, we treat all forces applied to it as being applied to its whole mass. When we model something as a system of particles, we model each force as being applied to a specific particle. The actual physical laws are the same in both cases, we're simply applying different models.
Note that treating a force as being applied to a particular point is itself a simplification used in particular models. A full quantum mechanical analysis would have the particle not localized to a point, so the force would not be localized to a point. In fact, in quantum mechanics, forces are themselves carried by particles, and those particles have some positional spread.