In accounting the second equation is the right one, thats because:
Equity (or owner's equity) is the owner's share of the assets of a business (assets can be owned by the owner or owed to external parties - debts).
Capital is the owner's investment of assets in a business.
The owner can also make profits from a business that he/she runs. These profits belong to the owner (they do not belong to anyone else, right?). Therefore, profits from a business are also part of equity.
Profits are kept in accounts called reserves.
Therefore equity consists of capital plus reserves (accumulated profits).
Capital is one sub-category of equity, reserves (accumulated profits) are another.
Hope that helps clarify the difference between equity and capital.
Assets = Liabilities + Equity. In this case, equity is residual interest. Capital is part of an eqity. See Warsono' paper (2018) superiority of the law of funds .... In the researchgate. Hope it's usefull.
I would like to offer a different perspective here:
Assets - Liabilities = Equity; my position is that business is separate from the owner thus A-L represent the business that is financed by E representing the interests of the owners (just an opinion) :)
Capital means money or funds. You can have a combination of common equity capital, preferred stock capital, and debt capital. Depending on the details specified in the preferred stock indenture this capital may be categorized as debt (liability) or equity. Given that, the first equation is not as refined as it should be and therefore is incorrect. The second equation (A = L + E) is correct.
In my opinion, equity is more comprehensive that capital. Capital can be considered in proprietorships and partnerships in which the owners are limited. Equity is more suitable for corporations with spread owners.