The prohibition has nothing to do with the cut itself. Scalp wounds heal quickly, and by 30 days (or 27) post injury, can be treated as though they never happened.
The issue is with the concussion, and to some extent, the strictness of the prohibition depends on the severity of the original injury. To diagnose a concussion, one should have either loss of consciousness at the time of the original event, or some neurologic abnormality on examination. Common ones include amnesia for the event, perseveration (repeating one's self), confusion, slurred speech, and the like. People who demonstrate neurologic abnormalities have, by definition, injury to the brain substance itself. This comes with an increased risk of recurrent symptoms, including seizures. In addition, a second head injury, however slight, can be fatal after a concussion. For this reason, sports activities are generally restricted for anything from two weeks to six months, until all symptoms have completely resolved (including headaches).
I would definitely talk to your doctor. You may not be able to talk to the doctor who treated you for the head injury, if you were treated in an ER, but your doctor should be able to access the ER records and see how severe the symptoms were that diagnosed your concussion. He can then explain to you the rationale for restricting your activities, and go over the risk/benefit ratio of liberalizing those restrictions early.
