How does the economy "detect" if money is removed from circulation?
Money is only inflationary in so far as it's moving around. So for there to be an inflationary or deflationary signal, that means that either the amount of money moving around has changed, or its velocity has changed.

And what does it mean to have money "moving around"? It means that it's being used to buy goods or services. And that's how it affects prices - because it affects demand.

So, yes, to elucidate: it's as you say- "if the millionaire wouldn't have destroyed the money but spent it, he could have bought a lot of goods with it, and the sellers of those goods could have bought still more goods with that profit and so on. As now those goods are no longer being bought, the sellers have to slightly reduce prices, so their goods aren't wasted in their warehouses. This means prices will drop a little, and this is how the economy "feels" the difference"