What has your client paid for so far?
If I were in your position, I would give the client the deliverables for whatever work he has already paid for, in its current state.
If some of the project parts that he wants are completed but NOT paid for yet --> then I'd explain that we can give him those parts for whatever additional cost would cover the work already completed on those elements.
I'd further explain that these deliverables are incomplete, however, and he's going to receive them in their unfinished form, reflecting the work that had been done up until his final payment.
I think that's the cleanest way to handle it without arguing over nomenclature, semantics, understanding of project manager process terminology, and standards, etc.
My own opinion, having been on both sides of this argument before:
- If I were your client, "cost-estimate" means an estimate for work not yet done. So I'd be upset if my contractor told me that he couldn't give me elements of work that I'd already paid for.
- From your perspective, I understand sometimes it's not always that clean and clear, but that's a risk PM's need to mitigate when these issues come up. (And similar things have happened many times for me. I've had lots of clients who were capricious and employers who couldn't say 'no' to requests that seemed overly complex and absurd to me as a PM. So sometimes you just need to do the best you can given the situation. This sounds like one of those times.)
As a matter of principle though -- I think clients should be able to get whatever work they've actually paid for, even if it's incomplete, and the project was canceled or not. Sometimes projects get messy, but that's okay. There are no hard and fast rules in project management (no matter what anyone says), and I think giving the client what he's paid for--and nothing more or less--is the fairest way to handle this case for everyone involved.
Hope it all works out.