When the Fed buys the bonds back, they remove the bonds from circulation, so it's an equivalent exchange. Likewise, when they sell their bonds as part of raising interest rates, they're removing money from circulation, which is also an equivalent exchange. The only time something is created from nothing is when the Treasury issues new bonds to fund increasing government spending.
Ok. So in that case, you're not counting bonds as "money". In that case, you can't count the stimulus as new money because that was funded by bonds.
I emphasize that this isn't a great interpretation because the root of the problem lies with Congress's inability to balance a budget, not the Fed. If our budget were balanced, then the Fed would instead raise interest rates by selling more bonds, which destroys more money. Because we're already issuing so many new bonds already, the Fed doesn't have to sell as many.
There are several definitions of what you can consider money. Physical cash is M0. Including money in your bank accounts gives you M1. In the modern age of electronic banking, M1 is what people broadly think of as money.
Treasury bonds are considered M4 money. It's not as liquid as M1, but so long as if you don't want to liquidate it, it isn't much different from M1.
>If someone issues debt and exchanges existing money for bonds, there is no change to the money supply.
I'm not sure what you mean by this. Issuing debt is creating money. This is true for commercial banks as well. When a bank issues lends you $10k to buy a car, that's money that had never existed before. You receive $10k from the bank. Meanwhile, the bank's depositors are still entitled to withdrawal the full amount they deposited in the bank.
When a government bond is first sold, the buyer gives the government money.
Then when the Fed buys it back, it creates news money for the purchase.