"The writer sat in the armchair and started writing."
Translation:Писатель сел в кресло и начал писать.
I agree that it's subtle but I think the fact that he "begins to write" shows that it's a change of state and not a description of what he's currently doing. Of course you might argue that he might have been sitting in the chair for a while and then suddenly started writing but without further clues that seems an unlikely scenario. I'm not a native English speaker but wouldn't you rather use the past progressive in this case? "He was sitting in the chair and started writing" for instance.
Perfective verbs describe a single moment in time and have no duration, so you can't "start" doing a perfective verb's action. (nor can you resume it, prolong it, end it, etc)
The imperfective is used here because you're describing the actual process of writing, without a goal specified, and that's exactly what imperfective verbs are for. If the sentence was, say, "I sat in the armchair and wrote down my statement." then you would use perfective because the relevant bit of information is the statement having been written down: «Я сел в кресло и написал заявление.»
Садился can be used to:
- describe that something happened during that action:
Он вспомнил о ней, когда садился в кресло - He remembered her when he was sitting down into the armchair.
- describe a repeated action:
Каждое утро пока все спали, он садился в кресло и начинал писать - Every morning while everybody were asleep he sat down into the armchair and started writing.