This might be clear to you at this point, but personal saving doesn’t just use the macro definition of saving. It uses both definitions. Personal saving has both basic components: a paper component, and a real production component.
The paper component will cancel out at higher levels, when you combine sectors, exactly as you say.
The real component will not cancel. At the macroest of macro levels, what remains will be the real component.
We can look at our last example. Economy 1: Personal income 100k from consumption good production, consumption 50k, exports 50k. Add to that Economy 2 with similar population and currency adjustment: Personal income 100k from investment good production, consumption 50k, imports 50k. Personal saving for both economies is 50k. For the first economy, this personal saving represents 50k acquisition of paper into the sector. For the second economy, it represents a negative 50k, from the loss of paper out of the sector, plus 100k creation of investment goods. When we combine sectors to create a macro-notion of “global saving”, the paper cancels and all we care about is the creation of investment goods.
But personal saving isn’t the whole picture, which is why the paper is still important.
Using the word “saving” contributes to popular misconception.
An individual is going to think of saving from a personal perspective: earning more than they spend. But that doesn’t work on a macro level, because income = outgo. Every dollar I earn is a dollar someone else spent, so it’s impossible for all of us together to earn more than is spent. But there is a deeper idea here, an economic dynamic that almost no one considers, of what we produce in order to prepare for the future. At the biggest level, the word “saving” gets assigned to this economic dynamic, the act of creating investment goods instead of consumption goods. Which is to say: saving is earning more than we consume. This is a vitally important idea. What’s potentially confusing is the label attached to the idea.
We might say that “personal saving” bridges the gap between the everyday idea and the deep macro idea, because it includes both the paper and the real component. The paper component cancels somewhere, sure, but the same is true when I as an individual earn more than I spend. People reading a news article on the topic won’t appreciate this double-definition. Hell, most people who write that news article won’t appreciate this. But there it is. There’s simply not an easy way to explain it.
As for boringness, well, I might agree with you but this thread has had about 80 views in the last 24 hours. I assume it’s not you refreshing dozens of times. I guess search engine bots would account for some of that, but the other possibility is that some of the people here are genuinely interested in the National Income and Product Accounts. One might think they had better things to do with their time. Like watching paint dry. Or finding what brand of mayonnaise tastes best with Doritos. Or giving serious consideration to what their hero name would be if they were a superpowered kitten that fights crime. (Mine is Avengakitty.) But they’re reading this. There’s something here that’s worthy of some attention. This is not a topic that comes up often. Or at all.
The big idea here is genuinely important, and worth some consideration, even if it’s a little too easy to get bogged down in extraneous details that hamper understanding.