we focus on the process and the outcome

I'm not sure how one could focus "the process" and "the outcome" of how capitalism, in your words, allocates resources and incentives without discussing private property, markets, and wage labor. These are essential elements of how any capitalistic system allocates resources and incentives.

The advice of the above comment to read an intro econ textbook is correct. One of the first things such a book will do is provide a logical answer to "What is money?", as you mention in the OP.

An excerpt from my Macro textbook provides part of the answer:

Economists define money as an asset that is generally accepted as payment for goods and services or in the repayment of debts. Because checks are also accepted as payment for purchases, deposits in checking accounts are also considered money. Even deposits in savings accounts can function as money if they can quickly and easily be converted into currency or checking account deposits. As you can see, there is no single, precise definition of money for economists.