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In Microeconomics, it is simple; capital is the means of production. Simple production functions have mainly two inputs: labour an capital. Capital is mostly fixed in the short run, and labour is more flexible. That is all well and good, but the problem arises in Macroeconomics, namely in the following identity: \begin{align} S-I \equiv X-M \end{align} where RHS is net exports, and LHS is called net capital outflow. What does that even mean? From what I understand, when there is a trade surplus, home has an increase in foreign currency, with which it will invest in foreign, either by holding the currency itself or investing in foreign assets, which for whatever reason means movement of assets out of the country.

I don't quite understand how that means assets are moving out of the country, but the bigger question is how does it relate to the capital in Microeconomics? They both seem to be completely different things, yet they are both called capital. Initially, I thought maybe they are just that; different things with the same name, but then Mankiw explains net capital outflow using the capital input in a Cobb-Douglas function. So there must be a link between the two I am missing.

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Capital in economics normally means machines and tools used in production. However, confusingly the national accounting uses the accounting terminology where capital just means:

Definition: Capital refers to the financial resources that businesses can use to fund their operations like cash, machinery, equipment and other resources (source).

The accounting definition includes all the things that economic definition does, but accountants consider also cash or other financial assets as capital.

If $X-M>0$ country experiences inflow of money. Foreigners have to draw down their reserves of our currency or purchase our currency on forex. If $X-M<0$ our country experiences outflow of money we have to draw on our reserves of foreign currency or purchase more of it on the forex market.

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