Basically the GP is using the former definition and the person you're replying to is using the latter definition.
Crap on the Marxists all you want but they do at least have a fairly unambiguous taxonomy for these distinctions.
This is why any of these definitions get really murky, fast.
On the other hand, someone who is worth, say, $10M or $20M is obviously still quite wealthy. But not necessarily in the doesn't need to think twice about hopping on a private plane or owning a private island category.
I think it is deeper than that, the country just has a confused relationship with the entire concept of class by both rejecting and embracing it.
I actually oriented myself more on the numbers for Germany. Which means middle class is a single household with about 2000€ net income per month. That includes a lot of trade workers. It is perfectly possible to finance a gap year without or just minor parental support.