When I first heard the term ~20 years ago, there was speculation that we would use up the easily extracted oil, and it would just get more and more expensive to produce as we were driven to more challenging sources, to the point that consumers would be driven to other energy sources. And so "Peak oil" referred to the peak of supply, as in the GP's usage. You are implying the peak will be whenever demand tops out, which technically would still be "peak oil" but not in the way it's been used for a couple of decades.

Peak oil implies that the demand actually reached a level to justify that peak level of extraction and with a decline[1] of the production that level of demand will be unsustainable. If peak oil is reached due to a temporary situation (like a war briefly driving up demand numbers) then maybe it's not an immediate issue - but we'd never have the same supply capacity again. It could be that in the 41st century earth is still producing 100 barrels of oil a year - but that's not a useful amount.

Running out isn't the issue - the issue is that we've got an economy geared to consume a specific fossil fuel and constantly growing with a dependency on that fossil fuel - if we suddenly outstrip supply we could be left in a lurch where we have a reduced capability to run the machines that'd let us build machines that are less reliant on oil.

1. The common understanding peak - but even if things just remained level supply-side and demand grows it'd be the same outcome

Peak oil was all about the supply side peaking, and what that would mean in an environment of increasing, very inelastic demand.

It looks like demand will peak first instead, which is quite different.

Short range fights might go electric. Long range fights may still use jet fuel, but hopefully jet fuel created using carbon from the atmosphere using renewable energy. That's carbon neutral.