The accounting term for this is "equity", not wealth.
I do not believe there is an accounting term for "wealth". I would be inclined to think of wealth as net worth above a prudent reserve, and above anything earmarked for a specific purpose. But that is a very hard line to draw.
For instance, a college fund for children or grandchildren, is not wealth. OTOH, savings for retirement can be wealth at some point. $10k of savings is not wealth, but $1 million is starting to be, and $5 million definitely is. And, it also depends where it is kept. There are a lot of descriptions for $1 million in a savings account, but wealth is not one that springs to my mind. Foolishness, an invitation to overspend, and a mark ripe for the plucking, all do. That same money as equity in rental property, as a balanced stock portfolio, as working capital in a small business, can all be described as "wealth in the making".
One of the problems here is that vanishingly few newspaper writers have ever had wealth, or even aspired to wealth. Therefore they use the term loosely.
That "aspired to wealth" does not mean sitting on one's rear end, cracking a beer, and saying "It sure would be nice to be a millionaire." It means developing a real financial plan to reaching that goal, sacrificing current consumption for future security, and actually starting to build wealth.
What the article is describing is "net worth", not wealth.
“Wealth” and “net worth” are roughly synonymous, or so they told me in my business-school days. The excess of assets over liabilities. In terms of your house specifically, that’s called “equity,” but over your whole set of assets and liabilities, it’s “wealth.”