They quote the textbooks accurately on banking, but that does not say that banking applies to what they were talking about. They might as well quote the Pythagorean theorem, which is true, but also not relevant.
So, that's the bidding. Now, time to play the hand. As I see it, the topic at hand calls not for bank accounting but for national income accounting expressed in terms of savings--sometimes called the flow of funds version of the accounting identities. For a closed economy (with no foreign sector), we have:
S - I = G - T
where S is private savings, I is private investment, G, is government spending ant T is taxes. The left hand side is net private savings, and the right hand side is the government budget deficit. As the government deficit goes up, net private savings necessarily increase. However, it is quite silly to say that the deficit is increasing savings, making it sound like, "Oh, boy, we're increasing our savings."
We have to remember that this is net private savings, which means savings minus investment. One way for net private savings to go up is for investment to go down. That is what is called crowding out, and it is not a good thing. As the deficit crowds out private investment, the nation's capital stock is reduced, which is bad for productivity and just about anything else that you would care about.
From the nation's point of view, the whole point of savings is to increase the capital stock. Any savings that goes to fund the government budget deficit is wasted from that perspective. We should not be happy about that.
For the next trick (hanging onto the bridge game metaphor), let us look at an open economy, with a foreign sector.
S - I = G - T + X - M
where X is exports and M is imports. Here, what we see is that all else equal, when net private savings falls, the trade balance falls (we run a larger trade deficit). Conversely, foreigners save more. See The Balance of Saving.
Phrasing it as Nugent and Mosler do ("U.S. consumers fund the desires of foreigners to save") fails to convey that our assets are going down while foreign-held assets are going up. It also fails to convey Ken Rogoff's point, which is that our consumption is crowding out not only our own domestic investment but investment in less-developed countries as well.
For Discussion. A few years ago, Argentina was by Mosler-Nugent logic adding to savings (i.e., running a budget deficit) and funding the desires of foreigners to save (i.e., running a trade deficit). How did that turn out?