In sectoral balance analyses, the 'government' is any part of state, regional or local government. So, when the government spends into another limb of government, the spending nets off against the cash asset and cancels out. It doesn't show up on the chart. The government has, if you'll forgive the analogy, moved a £10 note from one pocket to another and is neither in surplus or deficit in net terms.
This changes when the government money makes it to a 'household' or a 'business'. At this point the £10 note has left the government's pockets, and the government is in deficit and the Household/business is in surplus. Households and businesses are described as the 'private sector'.
For completeness, the 'trade balance' is represented. This is described as the 'external sector' or 'rest of the world'. There is always a net flow of British pounds in or out of the domestic economy. The British and US economies of the last 40 years have generally run trade deficits, and this shows on sectoral balance charts as an external sector surplus. This means that, in terms of financial value, UK or US residents bought more from external countries than those external countries bought from the UK or US. The corollary of this is that UK pounds have left the domestic economy, and are saved in the external sector somewhere.
In summary, the sectoral balances charts are a simple way to show a simple fact: every financial transaction has a buyer and a seller (or a borrower and a lender). So when money leaves the government sector, there will
always be a counterparty in either the 'private sector' or the 'external sector'. The same, in reverse, is true when the government taxes. Most sectoral balances charts look at a financial year (because the data are available) and they express the net position of each sector at the end of all the transactions in that period. A government deficit means that money has flowed to the private/external sector.