The example you post does not increase the money supply; if person A deposits 100 USD in bank A and it is lent to person B who deposits in is bank B, the money supply is still only 100 USD, and bank A would only earn the interest difference between interest charged to person B subtracted by interest paid to person A. This is not fractional reserve banking.
this is fractional reserve banking with a reserve rate of 0 %, so in a strict sense you're right, it is "no reserve banking".
the are many different measurements of money supply. the money supply you're thinking about is the 21 million coins, aka the "monetary base". you're right that bank lending does not increase the monetary base.
but bank lending doesnt increase the monetary base of current fiat monetary systems either.
so if someone says "banks create/print money" he is talking about a definition of money supply that includes bank deposits (usually M1 - M3 definitions).
the answer to that is: bitcoin banks would, too.
no difference.
I am not quite sure that I understand how a Bitcoin bank would operate. I am under the assumption that banks would lend Bitcoins and not USD; if this assuption is wrong, the following excercise is not valid.
Let us say that: a bank accepts deposits and receive 1000 Btc from various depositors. They give loans, but owing to the fact that they do not have Btc or rather the codes that Btc are made up from, they can never lend more than 1000 Btc. If the bank lends 1000 Btc no depositor can withdraw any Btc; which means that 1000 Btc equals 1000 Btc and not 2000 Btc.
Let us say that the depositors did recieve checks amounting to 900 Btc (100 Btc kept as a reserve). Business A receives a check of the sum 200 Btc. Naturally, Business A wants to withdraw the amount from the bank with immediate effect; business A has many liabilties in USD. If the bank has already lent
1000 Btc 900 Btc it cannot pay out the sum of 200 Btc which the check holder is claiming.
A more "sound" business model would be to loan 700 Btc which leaves 300 Btc for immediate withdrawal. In this case the bank could earn e.g. 6% on loans, equalling 42 Btc, while paying e.g. 3% on deposits, equalling 30 Btc, and earn 12 Btc. However, at no point would the monetary supply deviate from the monetary base.
Even with a reserve (not fractional) of 300 USD, it would not be possible to write checks amounting to 1000 USD with the depositors (which would in effect render the reserves "fractional") as business A would wipe out 67% of its reserves with a 200 Btc withdrawal, after which only a very small amount of depositors or other holders of checks could withdraw Btc.
If the Bank instead makes loans in USD, although being extremely risky due to fluctuating Btc value, the bank could lend 1700 USD (today's Btc price being 17 USD), or even 5000 USD, and any depositor could still withdraw Bitcoins. However, no sane person would operate such a bank as if Btc value were to rise to 25 USD, and all depositors wanted to withdraw the 1000 Btc deposited, the bank must purchase Btc for a sum of 2500 USD (or more), whereas the interest on the 1700 USD would only amount to 102 USD (with a lendning rate at 6%).