The first saving relationship comes from how income is defined:
Gross National Disposable Income - Household Final Consumption - Government Final Consumption = Gross Saving.So Gross Saving is most definitely Income that is not spent on Final Consumption.
The equivalent 'saving' figure for the external sector is called the Current External Balance.
In the National Accounts 'Net Saving' has a particular meaning, and refers to Gross Saving less the depreciation of fixed capital. This is not the same as the 'net saving' referred to in MMT, and it may be where some of the confusion creeps in.
If you go to the Capital Account, take Gross Saving and adjust for Capital Transfers between the sectors then you get the snappily titled 'Changes in Net Worth due to Savings and Capital Transfers'. (Clearly the national accounts people are running out of words too.)
Capital Transfers are investment grants and capital taxes like inheritance tax. Essentially it is a redistribution of value between the sectors and has no overall increase or decrease effect.
From the total 'Changes in Net Worth' you subtract Gross Capital Formation for the domestic private sector (which is the 'I' figure) and Gross Capital Formation for the government sector (part of 'G') and you will end up with the external sector balance (X - M).
The sector balances are called in the national accounts 'Net lending (or borrowing)'. This is the transfer of net-financial assets between the sectors. It is also the balancing item between the Capital Account and the Financial Account.
That is why sector balances are referred to as the 'accumulation of net financial assets' since on the Financial Account it is the balance between 'Total net acquisition of financial assets' and 'Total net acquisition of financial liabilities'. They are just different names for the same figure.
So for MMT in National Account Terms.
S is Gross Saving less Government Gross Saving = Gross Saving of the domestic private sector.
(S - I) is Total net acquisition of financial assets in the domestic private sector - Total net acquisition of financial liabilities in the domestic private sector. This is the Financial View.
(S - I) is also Gross Saving of the domestic private sector less Gross Capital Formation of the domestic private sector adjusted for Capital Transfers to the domestic private sector. This is the Capital View.
So if you are talking about saving now, you can be precise about which saving you are talking about.