My understanding is also that these discussions of "money" ignore things like investments or non-liquid assets, which I think is another big source of fuzziness. E.g. borrowing against other assets, including stock, that might have appreciated incredibly rapidly which gives you more purchasing power (the ability to "spend more money") without requiring anyone else to actually have given you money for anything specific.
This does not preclude, however, asset bubbles as we saw in Japan in 1991 and globally in 2008. The banks create loans which drives up the price of assets. Those assets, now appearing to be worth more, enable the banks to create bigger loans, because hey, the asset is worth more! This is a positive feedback loop and a major failure mode of this system. Regulation tries to tamp it down but does not always succeed.
It definitely is. When central banks think of the money supply, they take into account different aggregate which are sorted by liquidity.
The bit of information your second paragraph alludes to is the fact that all banks have accounts at the Federal Reserve. The Fed has the single database that the banks use to clear with each other. And the Fed and other regularity agencies audit the banks to make sure their internal databases are consistent, their loans are backed by assets of sufficient quality, etc.
This video by an economics professor is accessible to all and explains this to a certain extent in general. https://m.youtube.com/watch?v=4xgHbW2A9KE
Something to note is that in the US (and most modern economies), the federal government creates a 1:1 exchange rate between private bank money (e.g. money created through loans) and central bank money (numbers in the Fed database and physical cash) via deposit insurance (e.g. FDIC in the US).