Both of the Economist's concerns are well-known, but neither is uncontroversial.
Regarding "asset inflation," many people believe that stocks are too highly valued. Many others don't. Economics does not provide an answer to the question of who is correct in this controversy.
Similarly, it is common to express concern that Americans are spending more than they are saving. There are two possible holes in this concern: (a) The measures of saving we use are highly imperfect measures of what an economist would consider the be saving. (b) Economics does not tell us what the "correct" level of saving is.
Anyway, these two concerns do not add up to a conclusion that equities are not considered "savings" by economists. They most certainly do represent savings. If you earn $50,000 per year and spend $30,000 on food and shelter and $20,000 on stocks, you are saving 40% of your income. Putting your earnings into stocks or bonds or bank accounts are all forms of saving. Our official measures of saving, though, miss many important forms of saving. Consider the following: If you spend $30,000 on food and shelter and $20,000 on a graduate degree to help you earn income later, the official statistics count this as consumption, not investment or saving. If an American company owns an overseas factory whose value has doubled, this is really a form of savings, though the statistics might not recognize it as such. Why? The company's accounting may not require it to recognize this increase in value until such time as the factory is sold. Nevertheless, the increase in value is--to an economist--both income and savings.
Where layman's and economist's parlance do diverge is in whether equity purchases constitute "investment." If you earn $50,000 and spend $20,000 to build a small factory, economists would consider this an investment (and savings). If you spend the $20,000 to buy an existing share of stock, this counts as savings, but not as investment. While you might call it an investment, the economist would not. To an economist, an investment implies the creation of something of value. In this transaction, people have swapped things of value (money and stock), but nothing new has been created.
Note: Provided by EquilibriaChat, courtesy of the Federal Reserve Bank of Richmond. Please read their disclaimer.