Now given the above (you seem to agree with all the experts who say that, when accelerating, all parties agree that it is the accelerating party who is "really" moving), let's apply to a particular example.
Take a train, parked at a station, which suddenly begins to accelerate, at a VERY slow rate (say 1 mile per hour). After 60 hours it will be travelling 60 miles an hour. In the meantime, during those 60 hours, even SR will adopt an LR view of the deal. SR will say that the train is the one "really" moving during that time, and will therefore say that it is a clock on that train that is "really" running slow. In other words, according to BOTH a passenger on the train AND a stationary observer on earth, it is the train that is moving, and therefore both would assert that the train clock is the one running slower.
Now, the train stops accelerating, and moves at the uniform speed of 60 mph relative to the earth for one minute.
How would SR and LR treat that one minute differently? As we all know, the law of inertia says that a body in motion tends to stay in motion, unless it is acted upon by an outside force. So it would tell us that the train is STILL going 60 mph, even though it has now stopped accelerating. LR would say the exact same thing.
SR would NOT say that. In SR, the train passenger will now say (since it is moving at a uniform speed) the train is NOT moving, but that NOW it is the earth that is moving instead (again, according to a train passenger). In LR, the train passenger would concede that he is still moving, that's the difference.
To complete the example, let's now suppose that, after one minute of maintaining a uniform speed, the train begins slowing down at the same slow rate of 1 mph. So, at the end of another 60 hours, it will once again be motionless relative to the earth (it will have come to a complete stop). Here again, SR will concede that, during these 60 hours, it is the train, not the earth, that is relatively moving.
So after 120 hours and one minute, the train was, according to SR, "really moving" for 120 hours. For LR it would also be the one moving for the one minute--i.e, for the whole time.