A coincidence is something that happens to occasion remark, but which is, by definition, meaningless. That is, part of the metaphysical presupposition behind using the word 'coincidence' is that there is no meaning present. There is simply a factual occurrence which happens to provoke comment and interest in those perceiving it.
A Godincidence, by contrast, is something that happens to occasion remark but which is considered meaningful by the people involved. In other words, it is taken up into a larger story, that of their own life or the life of their community. One might also talk about providence.
This is not about proving one thing or another. This is about the assumptions embedded within the vocabulary.
Asserting that something is a coincidence - often accompanied with amplifiers like 'mere' or 'just a' - is the assertion of a specific metaphysical commitment, one which, in truth, rules out every possible sense of meaning (an unacknowledged consequence).
Part of the problem that Christianity faces is that this specific metaphysical commitment has not simply passed unnoticed, but that it has passed into the bloodstream of the church.
It needs to be extirpated. We might begin, as Christians, by disavowing the routine use of the word 'coincidence' and only using it when we are consciously asserting that there is no meaning to be found in an event.
I suspect that we would need to exercise a great deal of caution in such a case.