The banks emphatically do not get the money they loan out from deposits, because even though your bank makes loans, the money in your deposit is still there and available whenever you want.

Compare this to a private investment arrangement where after you invest (aka lend) your money, your money is gone (in exchange for the promise to get more back later, and/or certain other rights such as ownership in a company).

When banks lend out money, what they do is either give you an account with a positive balance (in this case, the money doesn't have to come from anywhere, because all the bank is doing is changing some rows in their databases) or they make a payment to another bank on your behalf. In the latter case, the money that you actually see as a bank customer again doesn't have to come from anywhere, because it gets created by the involved banks changing some rows in their databases. There is also settlement in reserves occuring behind the scenes, and if a bank's reserve position drops low, what I write elsewhere applies: As long as a bank is solvent, they can always get reserves either from another bank or, in the worst case, from the central bank.

As far as the money multiplier is concerned, there are two ways to look at it. The first is empirical: https://research.stlouisfed.org/fred2/series/MULT
How does a wildly varying money multiplier fit in with your story?

Also, there a countries where the reserve requirement is zero. The inverse of zero is not well-defined or infinity, depending on how you look at it. Yet those countries don't have an infinite money supply. How does that fit in with your story?

The second way is from first principles by actually looking at the laws. If you look into reserve requirements, you'll see that banks only have to satisfy them after the fact. That is, people in a bank's loan department first make loans, and then people in a different department of the bank go out and ensure that the bank satisfies the reserve requirements. They usually do so by lending and borrowing on the interbank market, but if they cannot borrow there, they can always get the required reserves from the central bank. See e.g. here: http://en.wikipedia.org/wiki/Discount_window

Finally, your last paragraph does show a potential causal pathway from increased savings to a higher demand for credit, via the interest rate. The import question is the relative strength of that pathway compared to others, such as the ones that I have described in my previous comment.

Obviously even economists end up disagreeing on that matter, given their political biases (though beware false prophets paid by wealthy people to spread lies and half-truths). Three final points on that particular matter:

(1) How sensitive to interest rates do you really think loan demand is? If a business sees growth in revenue, they are likely to invest and expand (which often involves taking loans) even in a high interest rate environment. On the other hand, if a business sees decreasing interest rates but also a fall in revenue, will they still take on loans in order to invest in a growth of their business? The answer can be yes occasionally, but it is more likely to be no. This is supported by surveys of business leaders.

This point is important, because in the face of high savings, businesses may see lower interest rates as a second-order effect, but they will also definitely see lower revenue as a first-order effect. When a large-scale shift of customer behavior towards savings happens, then businesses will see the fall in revenue first, long before banks lower their interest rates on loans.

(2) Empirically, some people have predicted for a long time, since the beginning of the financial crisis, that low interest rates would bring growth back via investment. And yet interest rates have been at the zero lower bound for some time with at best mixed results. (Again, the quantitative parts matter: Yes, low interest rates encourage demand for credit, but how strong is that effect compared to others that may go in the opposite direction?)

(3) The range in which interest rates actually vary by that mechanism is limited, because the central bank fixes the short term interest rate based on political considerations. (That's what e.g. the FOMC meetings in the US are all about. It's totally in your face, actually, but still many people refuse to really grok it ;-)) Obviously, the long term interest rate for loans taken by banks (which is some markup above that politically set rate) can vary, but the real story is control by the central bank.

[My comments are getting too long, so I stop here, even though there are more details to be talked about.]