4 definitions of money: all are correct?

Steve Roth has a piece in Evonomics:

Money makes the world go round. That may well be true, but money certainly makes the economics world go round. It’s the discipline’s special purview, the numeric linchpin that gives economics its dominant role and voice in our affairs. It’s what makes economics seem so “objective” compared to other social sciences.

Given that, it’s remarkable that economists don’t have an agreed-upon definition of the word. Sure, there’s the three- or sometimes four-part “money serves as…” non-definition that you learn in Econ 101. But in practice, what you see is a hodge-podge of unstated and shifting meanings in economic discussions.

The problem is, there are multiple, perfectly valid, widely-used meanings for the word. They’re all “correct.” The challenge is to be clear on each of those meanings, and on which ones are being used in a discussion.

Here’s a shot at that defining those commonly-used meanings, and the relationships between them:

money. n.
1. A  technology invented ca. fourth-millenium BCE for tallying up the value of diverse ownership claims, and designating prices, numerated in arbitrary units of account. “What’s the summed-up value of two goats plus three chickens?”

2. Wealth: the sum of tallied balance-sheet assets (see #1). Ask a real-estate tycoon or mutual-fund investor, “How much money do you have?”

3. Financial instruments whose market prices are institutionally pegged to a unit of account. Fixed-price instruments: physical cash, checking and money-market balances, etc. (The price of a dollar bill or a one-dollar checking-account balance is always one dollar.) A subset of #2. “How much ‘cash’ do you have in your portfolio, on your balance sheet — assets/instruments whose prices never change?”

4. Coins and currency. Physical tokens representing balance-sheet assets, which make it easy to transfer those assets from one balance sheet to another. A subset of #3. “How much money do you have in your pocket?”

He then takes you how when we say money in different contexts we mix all these definitions…

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