Obviously. Your fallacy is to go from there to assigning a state of motion to the events themselves.So the location where an event occurred is fixed, it has coordinates in that frame of reference
Again you fail to see the point: Events do not move. That they have some particular spatial coordinates in some frame is irrelevant. You could then argue that they would be at rest in all frames which would be absurd. Until you get this point you will not understand special or galilean relativity.A and B are fixed locations and cannot move relative to any observer in that observer's rest frame.
I teach relativity at university level. I am very familiar with the theory and I have seen your fallacy many times in many different people. If you doubt this you can ask any regular here about my understanding of relativity. You have failed to grasp a fundamental concept of relativity and until you dispell this misunderstanding you will struggle to understand relativity.Why do you have a difficulty with this? It is the fundamental basis of relativity - everything is relative.
And no, everything is not relative. In fact, as physicists we try to express as many things as possible in terms of invariants.