Is there any way to define coincidences so as to make their existence possible in a deterministic world?
I think so. Suppose you encounter an old acquaintance, whom you haven't thought about in years, on a street corner in a foreign city. That unexpected encounter sounds to me like a paradigm case of a coincidence, precisely because it was (as we say) "the last thing you were expecting." Nevertheless, the encounter might well have been guaranteed to occur by prior conditions, as determinism says all events are. Our very limited knowledge of the prior conditions -- indeed, our total lack of interest in their precise details -- makes such an encounter surprising, i.e., not at all predictable by us given how little we knew about the prior conditions. Even so, those prior conditions could have determined that the encounter would occur exactly when, where, and how it did.