I feel your confusion is somewhat justified by the lax way many textbooks talk about the subject.
Firstly, there is the reflexive pronoun sich in the third person. In a prototypical pairing like the following, the reflexive pronoun must be used to indicate that, as you would put it, someone is doing something to themselves and not to somebody or something else.
a) Eri liebt seinen BMWj und wäscht ihnj jede Woche.
b) Eri liebt seinen BMWj und wäscht sichi jede Woche.
The indices i, j indicate coreference; the reflexive pronoun must be used to indicate that the object is coreferent with the subject (b), a non-reflexive pronoun indicates that the object cannot be coreferent with the subject (a).
The verb waschen, however, is a perfectly ordinary transitive verb. What does ordinary mean? In linguistics, one would use a lexical entry like the following:
Semantic arguments: agent (who is washing?), patient (who or what is being washed?)
Syntactic arguments: subject, accusative object
Linking: agent – subject, patient – accusative object
Transitive verbs select for two arguments, a subject and an accusative object, with the subject interpreted as the agent (the person doing something) and the object interpreted as the patient (the person or thing something is being done to). For further reading, see the Wikipedia entry for argument.
Secondly, there are verbs that are truly reflexive (echt or obligatorisch reflexiv in German). They are distinguished by the fact that they have a semantically empty syntactic argument, i.e. one which is not linked to a semantic argument.
Semantic arguments: agent (who is hurrying?)
Syntactic arguments: subject, accusative object: reflexive
Linking: agent – subject
What this means is that in a sentence like the following, there is only one person doing something (as indicated by the English translation), despite the fact that there is an object present. The object has to be coreferent with the subject – reflexive – i.e. in the third person, it must be sich.
Er beeilt sich.
Finally, note that there are some verbs that are truly reflexive only in some meanings.
Der Geist beherrscht den Körper. (transitive: Geist = agent, Körper = patient)
The mind controls the body.
Er beherrschte sich. (truly reflexive: er = agent)
He held his temper.
Sie übergab den Schlüssel. (transitive: sie = agent, Schlüssel = patient)
She handed over the key.
Sie übergab sich. (truly reflexive: sie = agent)
The confusion arises because textbooks usually introduce reflexive pronouns with ordinary transitive verbs like waschen, but do not make the effort to explain what a truly reflexive verb is. In the mind of the student, reflexivity becomes equated with sich and the two cases er wäscht sich and er beeilt sich are not clearly distinguished.