I don't understand this M jargon - but I fear that is old economics! Still I suggest that you recognise that although Building Societies do, Banks do not transfer money in any way whatsoever! I have to say again: Look at the Bank of England link. Private banks create money every time they create a loan. Indeed that is effectively the definition of a bank. If you want another justification then here:
http://www.sciencedirect.com/science/article/pii/S1057521914001434?np=y
For M0 to M4:
http://lexicon.ft.com/Term?term=M0,-M1,-M2,-M3,-M4
Different measures of money supply. Not all of them are widely used and the exact classifications depend on the country. M0 and M1, also called narrow money, normally include coins and notes in circulation and other money equivalents that are easily convertible into cash. M2 includes M1 plus short-term time deposits in banks and 24-hour money market funds. M3 includes M2 plus longer-term time deposits and money market funds with more than 24-hour maturity. The exact definitions of the three measures depend on the country. M4 includes M3 plus other deposits. The term broad money is used to describe M2, M3 or M4, depending on the local practice.
As a thought excecise -- it's been nearly 20 years since I did GCSE economics.
Bob has £100 and puts it in HSBC (HSBC has £100 cash inside it)
HSBC loans £90 to Carol and wants £10 interest (HSBC has £10 inside it and is owed £100 from Carol and owes Bob £100 - total £10 surplus)
Carol buys a car from Dave for £90
Dave puts £90 in his bank (HSBC has £100 inside it, but owes Bob and Dave a total of £190, and is owed £100 by Carol)
HSBC loans £80 out to Eric and wants £10 interest (HSBC has £20 inside it)
Eric pays Carol £80 for a ride to the station
Carol pays back £80 of the loan with HSBC (HSBC has £100 inside it, owes Bob and Dave £190, and is owed £20 by Carol and £90 by eric, so a surplus of £20)
Dave takes out £90 and pays Eric for a juggling performance. Eric repays his debt (HSBC £100 in cash, £20 in debt (from Carol). It owes £100 (to Bob)
Bob takes £20 out and pays Carol for gardening. Carol pays off her debt. (HSBC has £100 in cash, not owed anything, but owes £80 to Bob)
Bob takes £80 out and keeps it under the bed.(HSBC has £20 in cash, not owed anything, doesn't owe anyone)
Now al debts are paid and there's still £100 in the economy, it's just that a portion of the original £100 (£20) has been transferred to the Bank.
Half way through this game, it looks like HSBC has created an extra £90 (Bob and Dave have a total of £190 in their current accounts), but that money is backed by the debt of Carol and Eric, and once all debts are paid we see there's only really £100 in existence.