Other than the reserve portion going to the Federal Reserve (it can also be kept at the bank itself or at the Fed or some combination of both) this IS how it works.Repeatedly posting this fictional account does not make it any more accurate.
I deposit $100. I no longer have $100- the bank does. They give me a promise to pay later which is considered a liability for the bank. With a ten percent reserve requirement, the bank can loan out up to $90 of that. Somebody will owe them the $90 so that is counted as an asset and the $10 on reserve is an asset. Now they have $10 and the borrower has $90. If that person A does not need the money at that moment, he can deposit his $90 with the bank. He no longer has $90 but a promise from the bank that they will pay him back his $90 later. On this deposit, the bank can loan out $81 and has to keep $9. Now the bank has $11 and person B has $89. Everybody else has a promise to be paid back.
Now if I want my $100 back, the bank has to get that money from someplace. They can't make it from nothing. They can get money from a new depositor or from another bank which may have more reserves on hand than required for the amount of loans outstanding or they can borrow from the Federal Reserve until they can attract or borrow more money from some place else.
What has also happend is that the bank no longer has the $100 deposit they needed to be allowed to have that $90 loan out. They have to keep $10 on hand in reserves to keep that loan out so they also have to get another $10 for reserves.