"The cook starts in October."
Translation:A szakács októberben kezd.
A few more pairs like this:
"játszik" and "játszódik" - "play" and "takes place" (a story, a play, a film)
"befejez" and "befejeződik" - "finish" and "ends" (a story, a film, a book)
"fokoz" and "fokozódik" - "increase/intensify" and "intensifies" (noise, the strength of something)
I don't think there are too many of these, but I'm not sure.
There are, simply, "-ik" type verbs ("ikes igék"), the conjugation of which is a bit different from that of other verbs. The most characteristic difference is the "-ik" ending in the present tense indefinite third person. And, usually, the present indefinite first person singular ends in "-m", not the usual "-k".
But this all does not depend on anything. This is simply how these verbs are conjugated.
The verb pairs I mentioned above are separate, distinct verbs, even though derived from the same root. One of them just happens to be an "-ik" type verb, because that is how they were created.
Those passive/reflexive(? Sorry, I don't know the correct tem) verbs are frequently like this, "-ik" verbs. But not always. For example:
"indul" means "starts", the intransitive meaning of the verb.
"A busz indul." - "The bus starts."
The subject itself is the receiver of the action.
The transitive meaning is carried by another verb:
"indít". "A sofőr indítja a buszt." - "The driver starts the bus."
Of course, "indít" and "indul" are also relatives, but they were derived differently, neither of them is an "-ik" verb.
Anyway, all these verbs are separate, individual verbs in their own right. But it can be difficult for sure for English speakers because they can translate to the transitive and intransitive meanings of the same verb in English.