> That only works if the government buys something that no one else wants; in which case it is buying at a price, ipso facto, that no one else will pay (at that price which is above the market price).

Not at all.

Suppose the government funds research. Private entities fund research too. It's clearly worth something. But if the government funds it, the research happens in their jurisdiction. Even if the exact same private company paying the taxes that fund the research might have done the research themselves, they might have done it somewhere else, so the government is now creating a subsidy for doing research in their jurisdiction.

The value of the research could be fully identical regardless of where it happens, but the government subsidizes it because they want it to happen there.

> But if the government buys something at the same price (or lower) that everyone else pays, then it isn't a subsidy.

It could only not be a subsidy if the thing they're buying is identical in all respects to the thing the taxpayer would have bought had they been left to keep the money. Otherwise it's subsidizing the thing the money is being allocated to over the thing it would have been allocated to. That's what subsidies are -- the reallocation of resources through action of law. It's a synonym for spending tax money.