You're confusing private banks with central banks. I don't think anyone is surprised that the U.S. Fed reserve regulates the money supply in part by creating it.
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.
Everyone's a bit confused. Or talking at cross purposes.
* Central banks create M0 (either paper currency, or electronic equivalents - original fiat money). (I had to say fiat, or some gold bug would complain that gold is the real money).
* Banks borrow money, lend it out again, then when the money comes back to the banks they keep recyling it in an infinite cycle, creating M0/fr amount of broad money (M2?).
* Then other infinite cycles create even broader money, but let's forget about that.
A lot of people (especially bitcoin and gold proponents) forget is that the government can still create M0 if it's not just greenbacks (by lending out "imaginary" gold or bitcoins - as long as someone will take an IOU from the government in return for a little bit of interest). Banks can create M2, as long as someone will take their IOUs (with interest). You can ban lending, but since the critics of money usually lean Libertarian, that's not an argument you often get.
What some post-keynesians and some economically literate marxists (and maybe MMTists, I don't actually know what they're on about) suggest is that the fraction can suddenly change.
Boom - lowering fractional reserves, which means the total amount of broad money keeps increasing.
Bust - banks try to lower fractional reserves, which makes the total amount of broad money contracts.
The fraction is often treated as a constant, but if it suddenly changes (or the desired target suddenly changes), it can have a massive impact.
Banks get very comfy lending out money with very low reserve rates (effectively the multiplier of money), because whenever there's a crisis the government will "print" M0, and hand it over. So as long as the banks don't act more irresponsibly than every other bank, they can't fail. "Macroprudential" reforms (telling banks to stop it when they lend to much, or risk fines / penalties) is becoming fashionable for this very reason, since telling banks they'll all go bankrupt is about as credible as a teacher yelling "If the class isn't quiet I'll expel the lot of you!".
I like to consider the human side - in the asset bubble that occurs when M2 is rapidly growing, everyone wants to work hard and save hard, because otherwise they'll never get on the property ladder. That creates a boom, then a bust when they realise it's a Ponzi scheme. Economists who know better talk about high investment, then a crash in investment when the money suddenly dries up.
(See analogy above, do not assume I see this as good educational practise)