It's the nature of this kind of question to get a lot of answers focusing on the differences. So to counterbalance that: I'd like to tell you that these words are, in fact, very close synonyms.
When they are used with their literal senses to refer to people, there is no simple distinction in what they denote like the one you read online. For most people, the idea of a contrast between "the rich" and "the wealthy" would seem to be an oxymoron.
That doesn't necessarily mean these words are always interchangeable, but the factors governing which can be used are more complicated than just a matter of easily-described distinct meanings. Similarly, while there are differences in connotation, neither one will always be taken to be more appropriate/upper-class than the other.
The source you read, Rich or Wealthy (an editorial by
Tom Wheelwright), does define these as referring to two distinct types of people. But this definition, even though it might look like a normal dictionary definition, is not actually a description of how people use these words. It's a rhetorical definition. The author wants to describe some significant difference he sees between two types of people who are normally classified in one group. To describe this difference, he resorts to taking two words that are normally synonymous, and more-or-less inventing a distinction between them that corresponds to the distinction that he wants to talk about. After all, the introductory sentence—"
It seems to me that there is a big difference between being rich and being wealthy"—would not make for a very strong "hook" if all native speakers distinguished the meanings of these words in actual usage. It's supposed to make readers think "What? These words are synonyms! There is no big difference between them!" and want to keep on reading to learn the supposed difference between them.
Differences in actual usage
"Rich" is more common than "wealthy"
Here's the Google Ngram Viewer chart:
This suggests that in general, if you're unsure which word would be better, you should go for "rich".
"Rich" has more general meanings that can apply to inanimate nouns
As Mick mentioned, one of the clearest actually observed distinctions is that "rich" can be applied to foods, or other inanimate objects, in ways that "wealthy" can't. "Rich" foods are those that are very filling, dense or fatty (possibly overly so); we can't say "wealthy foods". Another metaphorical use of "rich", applied to inanimate nouns, is to mean something like "abundant" or "fertile": you can say something like "the artist's painting was rich in meaning".
In some colloquial contexts, especially in predicate position, "wealthy" sounds bad
I need to research this more, since I can't explain it very well yet. But as an example, "I'm gonna be rich!" sounds much more natural than "I'm gonna be wealthy". Note that this does not mean that "rich" is only used in colloquial speech.
Supposed/proposed more philosophical distinctions
wealthy = "know how to make and keep money" vs. rich = "just has money"
You mention that idea that “rich people only have money while wealthy people in addition know how to make money”. I think there's some small kernel of truth to the distinction made in the article you read, but it's not observed by everyone.
I think these words can sometimes imply this, but it's not a core feature of how these words are defined. Its an additional implication that exists in some contexts, or for some speakers.
I think this definition seems like how people who consider themselves “wealthy” but not “rich” would define the terms. Note that this definition relies on a worldview where some people “know” how to make money (rather than having money because of inheritance, luck, or other external circumstances), so you’re especially likely to encounter definitions like this in contexts where people want to distinguish their “how-to-become-wealthy” advice as more legitimate than “get-rich-quick” advice.
My issue with this definition is as follows. When a speaker describes someone else using the adjective “wealthy”, I don’t think it’s safe to assume the speaker means to imply “that person knows how to make money”. In my experience, the speaker may just mean “that person has a lot of money”.
“Not one member attended Anne Henderson’s wedding to von Korber, and
the once wealthy heiress was destined to spend a life that at times
fringed on poverty.” (Canadians in Golden Age Hollywood: 2-Book Bundle, by Charles Foster)
In this sentence, I don’t think the author meant to imply that Anne Henderson used to know how to make money, but later in her life somehow lost this knowledge. It just means she used to have a lot of money, and then lost most of it.
"rich" = high-income vs. "wealthy" = possess much wealth/assets
Actually, I found another source that gives a different, although in some ways similar definition.
The Graduate Student's Question: Before the Last Tree, by Walter L. Battaglia, says
The important distinction between the rich and wealthy is this: "rich" usually refers to income while "wealthy" refers to accumulation or property. Thus, to be rich is to have a high income, whether or not one possesses anything. To be wealthy is to be in possession, whether or not those holdings produce income. [...] The wealthy were always perceived as more secure than the rich, as long as they didn't waste their Estates. The rich can be quickly deprived of income. (p. 240)
I said "in some ways similar" because this account seems to agree with Wheelwright about "wealthy" being a more secure state than "rich". But it contrasts with Wheelwright about what the state of being "wealthy" actually looks like. Wheelwright paints a picture where, whatever assets you start out with, you can achieve this stability with the right attitude and know-how:
Once you know how to make money, you can build sustainable wealth. The money never stops coming. If you have a reversal of fortune, it's not a big deal. You just make it back.
And contrastingly, people who just become "rich" through inheritance are, according to Wheelwright, not necessarily "wealthy" because they may squander their wealth.
Battaglia on the other hand seems to have the viewpoint that possessing property, even if it doesn't guarantee security, will still naturally confer more security than just getting a high paycheck.
In general, people inclined to draw fine distinctions seem to feel that it is proper to use "wealthy" to denote a more stable state than "rich", which is seen as implying a more "easy-come, easy-go" state of affairs. However, there doesn't seem to be a consensus on the exact nature of the difference. Some people think it is appropriate to use "wealthy" to refer to people with the stability supposedly conferred by the possession of lots of property, while others think it is appropriate to use "wealthy" to refer to people who possess personal qualities or virtues that supposedly enable them to consistently do well financially.
A minor note on parts of speech
By the way, I believe that many modern linguists would say neither of these word are ever nouns. That's the kind of conflation of word class and grammatical function that dictionaries often exhibit. "The wealthy" and "the rich", like "the poor", can be used as plural noun phrases, but the words are still adjectives in this context. One piece of evidence for this is that despite being plural, they don't take the regular nominal plural suffix -(e)s. A more detailed argument against interpreting these words as nouns can be found in the following paper: "The rich, the poor, the obvious: Arguing for an ellipsis analysis of 'adjectives used as nouns'", by Christine Günther.
To be fair, Günther does say that other linguists have analyzed such cases as conversion from adjective to noun, so the dictionaries could just be considered to be following an old-fashioned analysis. Here's another overview of the matter I found on Google Books in "Hectic, Hippic and Hygienic: Adjectives in Victorian Fiction: A Semantic Analysis" by Chris Kunze.