Is the word sin an archer's term?

Yes it is, it means whenever you didn't hit gold (the yellow rings in the centre) during Archery practice or during a competition.

The church's version of "sin" comes from the idea of a sin being an imperfection, a flaw, that can be improved on, and any Englishman during the entire medieval period could relate to this idea.

(NOTE: I am a real archer and am an expert on the topic)