A bank's capital is used to finance the purchase of assets. Those assets are mostly loans made to households and businesses and governments. When a loan is made the bank is "buying" the future payments of the borrower. This suggests the whole right side of the balance sheet is capital because it takes the entire right side to finance the purchase of all of the left side. Some of the capital is the bank's own debt or equivalently, the liabilities, which is owed to depositors and bond holders. The remainder of the capital is called equity which is what the common and preferred shareholders own. The value of all the shares or of all the "stock" (as in stock market) is the equity. Stockholders and shareholders and equity owners are the same thing. Confusingly "the stock in a business" is the assets (in blue on the left) but the "stock" (also known as the shares) are the smaller part called equity (in teal on the right). In order to control all the assets (in blue) you have to own the equity (in teal) so that is why the word "stock" is used to refer to the teal part and also the blue part.
Net worth is assets minus liabilities so it is the same thing as equity. "Net worth" is the usual term when speaking of the equity of an individual or a household. When speaking of the equity of a company sometimes stock market participants call it "book value".
I don't think non-bank companies think of these terms any differently than banks. In any company there is the distinction between physical capital which is always on the left side, and financial capital which are the two parts on the right side. If the company has some cash or other valuable paper on hand then there is also some financial capital on the left side (held as an asset).
The above diagram is useful in the context of accounting and finance but less useful in the context of economics.
I think economists use the word capital to refer to an accumulation of value, in other words assets. In economics capital has the connotation of being an accumulation rather than a flow. For example savings are an accumulation of a flow of income. (Capital) accumulations can be land, structures, equipment, software, patents, copyrights, and human capital (skilled people and educated people), and other assets.