Paper money is clearly not the same thing as an electronic record in a bank account. Paper can't be stored in a database. They are kept equivalent because banks (and people) trade them at par, for example using ATM's.
Each bank (including central banks) has its own computer system for its accounts. Bank account money never leaves a bank's computers. There's no way to get it out of the computer, any more than you can remove your virtual treasure from an online role playing game.
So this seems like a good way of thinking about what happens when a bank creates money. They can create virtual currency in their own computer system, not in anyone else's. It doesn't make them richer, any more than a game company creating virtual gold pieces makes them richer. The money is either meaningless (if held by the bank itself) or a liability (if it belongs to a bank customer). To a bank, only outside money counts as wealth.
Transfers happen via trades. To pay anyone not using the same computer system, a bank needs outside money of some form.
So inside money and outside money are clearly different. To a customer, money in Bank A might seem equivalent to money in Bank B, but to Bank A, only dollars in Bank B are assets, and to Bank B, only dollars in Bank A are assets.
So, one way to approach the "what is money" question might be to look at payment systems. How is it that all these different sub-currencies are made to trade at par?