The Federal Reserve, on the other hand, can create money from nothing and use that to buy government bonds, like they did with QE.
it affects inflation, if not as much as M0 does iirc. (its been a while since I've taken econ, so i'm a little fuzzy on that)
This all works fine assuming you don't try to withdraw all $100 since they only have $20 on hand. Nor would it be good if Erin defaults on her debt. Or the bank collapses. This is what FDIC is there to protect, where the government will create new money in case things collapse.
And what will Erin do with that money? She will give it to Tony in exchange for some service. What will Tony do? Pay that into his bank account. Which means the bank now has an extra $70 (assuming they keep a $10 reserve) that it can loan out, bringing the total amount of money in our economy to $250.
Also that $80 that Erin borrows will need to be paid back with interest. Say the interest is $20. She pays back $100. The bank now has $120. That $20 that she paid back had to come from somewhere. Say she sold a blanket she knitted to Paul to make that extra $20. Where did Paul get that money from? He had to borrow it from the bank. And the cycle continues...
Yeah but Erin had to put his house for getting his loan.
Private banks aren't creating money, the central bank is. You've confused the two.
Which brings me to your own "M2 is derived from M0". There are certainly interpretations of that statement with which I would agree, but there are also interpretations which are seriously dubious. For example, the interpretation that "there is a causal mechanism from the quantity of M0 to the quantity of M2" is highly doubtful. The economists who actually study those things seem to come down mostly on the side of endogenous money, which says that the causality is the other way around: "changes in M2 drive changes in M0" - or rather, they did before interest rates went to the zero lower bound, at which point the two quantities became decoupled.