Uh-oh, you DEFINITELY need to take Macro-Economics in college. (Your comment reveals you are "in over your head.")
1) I am not simplifying anything. I gave you the ONLY definition of economic "recession." The ONLY thing that determines a recession is quarterly GDP growth.
2) GDP (and, thus, "GDP growth") includes far more than the government sector. It also includes consumer spending, business investment, and net imports. So, GDP includes ALL sectors, including agriculture, manufacturing, wholesaling, retailing, services, mining, and transportation.
3) According to Keynes, government fiscal policy is a perfectly valid tool available to the goverment to maintain and stimulate the economy. Government spending, bostered by the "multiplier" (learn about this in your Macro-Economics class), often obtained by government borrowing, does not "dig holes." Government spending helps/stimulates the economy. (Of course, the "trick" to government fiscal policy is: governments should run deficits (i.e., spend) when the economy contracts but should revert to running surpluses when the economy expands. Between Reagan (who tripled the national debt) and Bush/Cheney (who quadrupled the national debt), only Clinton ran surpluses (sadly, only for two years).)