I think production peak oil implies prices will rise as the market adjusts, while demand peak oil implies the opposite.

No, it implies the exact same thing - prices will rise as the market adjusts (and "economies of scale" start to crack).

That's not how the economics of mineral resources work. The marginal cost is increasing with volume, because the resources become harder and harder to find. Economies of scale only operate on the scale of individual extraction operations, which individually are vastly smaller than the overall market. That is, if demand goes down, it will not be that the same extraction operations are run at half speed, losing economies of scale. Instead, the most expensive such operations will simply never be started, and the price will fall to reflect the new cheaper marginal cost.