In Economics, to define/describe "demand for $X$", whatever $X$ represents, we need
- a definition for $X$
- a measurement unit for the quantity of $X$
- a measurement unit for the price of $X$
Given these, the "demand schedule" for $X$ (terminology dating back to A. Marshall), is the geometric locus of pairs (quantity, price) that reflect the quantity demanded for the corresponding price in the pair.
And that's that: why is the quantity demanded what it is for a given price, is (at least) one level deeper in the economic structure and analysis.
As regards "money", I just record here the usual approach taken:
- We have in mind something that is perfectly liquid (and in practice we may include in the definition "essentially liquid" assets), where "liquid" means "immediately acceptable by anyone in any exchange"
- The quantity of money is measured in some currency unit (like USD, Euros etc),
- We use the interest rate as "the price of money"
To go this one step deeper, the reason why "demand for money" is not fixed at infinity, has to do with the question that we pose so as to avoid a useless answer. And the question is essentially a question of allocating a given level of assets between different forms/different degrees of liquidity, money included: Given that my wealth is $W$ (measured in a currency unit), and given that the interest rate I will enjoy if I save it all in an interest-bearing form is $i$ (unitless), how much of this wealth do I want to keep in "money form"?
Enter arguments as to why we want to keep money, functional forms that formalize these arguments, etc.
In dynamic models, it is understood that this question is posed and answered by the demand schedule for money for each time-period of the model. In static models, there is a single time period.
Another usual aspect is that we examine "real money balances", i.e. the deflated value of the nominal money balances.
"Current holdings of money" is not the demand schedule but equilibrium quantity demanded, given whatever constraints we postulate that they exist.
This gives rise to the concept of a possibly "liquidity constrained" economic agent: given his wealth, the interest rate and his preferences, the agent would want to allocate $W_m^*$ of his wealth to money holdings. But we may observe that it allocates something less, for example because the existing markets where the agent can transform non-liquid assets to liquid ones are "incomplete" (as they say), and so the agent finds himself with non-liquid wealth that he would want to transform into money but he cannot (perhaps due to asymmetric information, risks that cannot be priced, etc). While this is a constrained equilibrium also, it is important for economic analysis and economic outcomes to realize that here we may have an additional "binding constraint" on the agent's behavior, and as this constraint changes, the behavior of the agent will also change.
Detecting the existence of such exogenous liquidity constraints is difficult, because it is difficult to separate their effects on money holdings from the effects of other constraints (like the level of existing wealth of the agent).
So "current money holdings" is tautologically equal to "current equilibrium money demanded", but the challenge of economic analysis is to determine which constraints are binding and so characterize the observed equilibrium.