I didn't intend to start some kind of No True Scotsman inspired argument about who's in and who's out of the ERE club

My original point was that not all networths are created equal. It's just a nice summary but it doesn't really reveal the particulars of someone's financial status.
Consider, e.g. Mark Zuckerberg whose NW (48 billion) is mostly tied up in facebook stock. His fortune stands and rests with how FB performs. On top of that he can't convert that FB stock into cash and expect to end up with the same amount of NW as he nominally has now. As soon as he starts selling, FB price will drop ... likely he won't be able to find enough sellers to absorb all the shares he holds. It's not unlikely that maybe 30 billion in cash (or treasuries) is worth more than 48 billion in FB shares if the goal was to actually spend the money on something else.
Compare Zuckerberg to old-money wealth. Here a lot of the wealth is tied into an intricate web of cash flows and possessions. In many cases it's even hard to estimate just what the NW is. However, old-money becomes old because it's quite robust. Such a person always has some money coming in from somewhere. They're not just the founder and CEO of one company. They own many companies. They're on the board of multiple companies. They have well-connected friends. They own politicians (local or national).
I'd somewhat compare the different kinds of wealth both to investment methods, gardening methods, and FIRE strategies.
Level 1: The garden only grows potatoes. Success rests on how the potato harvest turns on. In terms of investing, it's like having your money in a single issue. In terms of FIRE, it's like a consumer with a lot of money. Representative billionaire: Mark Zuckerberg.
Level 2: The garden contains of random rows of different plants. The rows are diversified. If beans fail, maybe potatoes still do okay. It could be that the entire harvest fails if nothing was planted for a surprise drought. In terms of investing, this is like diversification. In terms of FIRE, it's the person with a side-gig or three and a bag of tricks to save money. Representative billionaire: Warren Buffett.
Level 3: The garden contains non-random guilds where things are planted to support each other and energy flows are deliberate managed. In terms of investing, this is like a hedge fund. In terms of FIRE, it's the web of goals approach to advanced (deliberate side-effect, reverse fishbone) contingency planning. Representative billionaires: Koch brothers.
The higher the level, the more robust/resilient the strategy is and the less money is likely needed.
It's probably a bad idea to rank levels ... but it seems pretty obvious that these three approaches are quite different from each other.