Duden gives the following examples for the use of verletzen:
ich habe mich am Kopf verletzt
ich habe mir das Bein verletzt
Why is the reflexive pronoun accusative in the first and dative in the second? And are the following variations valid, and, if so, how do the meanings differ from the alternative:
ich habe mir am Kopf verletzt
ich habe mich das Bein verletzt
The difference lies in whether the accusative object tells a body part or a person.
Ich habe mich am Kopf verletzt.
This tells who is injured. The location of the injury in told as an adverbial
Ich habe mir das Bein verletzt.
This tells what is injured. Instead of a location of the injury, who bears the result of the action is told as an addition. As usual as a dative object.
The latter form is also useful if you want to tell where it happened.
Ich habe mir im Urlaub das Bein verletzt.
Ich habe mir am Kopf verletzt.
This is ungrammatical because the verb verletzen requires an accusative object. People who hear that will likely ask Was hast du dir verletzt? asking for the body part they overheared.
Ich habe mir meinen Kopf am Kopf verletzt.
Ach! Du meinst deinen Kopf am Kopf der Fräsmaschine. Tja. Nächstes mal besser aufpassen!
Ich habe mich das Bein verletzt.
This is ungrammatical because it features two accusative objects and the verb verletzen only takes one. People will likely correct you Mir. Du meinst mir.
German dative is a bit unique in that it has carried over some notions from Latin that don't appear in other European languages. One of them is what's called the dativus (in)commodi - this construct uses the dative to denote the object that experiences the advantage (or harm) of the action (in your cases: "mir")- so a somewhat "second accusative use".
So, your examples
ich habe mich am Kopf verletzt
and
ich habe mir den Kopf verletzt
are somewhat equivalent, and both translate to "I hurt my head". Note your accusative example needs a preposition "am" to denote the body part that was hurt, the dative doesn't, because it carries the notion of the preposition. That is also the reason why your second set of examples (where you use the dative with a preposition and the accusative without) are wrong.
Transforming the sentence to "I gave myself a hit to the head" (note the dative in English) might somewhat explain how the German dative is used here.
Janka's answer already contains the point, but it is somewhat hidden: When you say "ich habe mich am Kopf verletzt", you basically say: "ich habe mich verletzt" and, specifically, "am Kopf". In contrast when you say "ich habe mir den Kopf verletzt" you have two objects (dative and accusative), which somehow interact with each other, and neither can be omitted.
Likewise with Carsten's comment: "Er schlägt ihn" und zwar "ins Gesicht". One object. "Er schlägt ihm ins Gesicht" = two obligatory complements (one is a prepositional phrase).
The difference in the content is: When the person is the accusative object instead of the body part, you indicate that the person as a whole is affected (and that's the main thing, the specific body part can then be omitted).
