Open source used to be used as a marketing tool --- people believed that software was automatically better if it was Open Source. Today, people have learned how to be more nuanced; just because it is better in some FSF sense (if it is "free software" in the Stallman/GPL ethical/moral sense of the word) doesn't mean it's better for a particular business purpose. If you're a developer, or your company has a development shop where it's prepared to make changes, then obviously Open Source is better.
If however, you're prepared to contract out support and development to a company like Red Hat, then from your perspective as a customer, it may not matter as much. As long as you are willing to pay $$$ to Red Hat and the software/services/support is of sufficient quality for your purposes, you're happy. Now replace "Red Hat" with "Oracle" in the above statement; does it still work? Why, yes. So you really won't care if it's Open Source or not.
At one point there was an argument that Open Source is good because you're free to change to a different vendor. i.e., if you don't like Red Hat's services, you could always switch to SuSE. In practice, alas, the ways in which various boot and configuration scripts are set up are just a tiny bit different enough that it's often not trivial to switch from one Linux provider to another. So the main difference is whether or not you're willing to make changes to the source code and recompile. But if you're a Red Hat customer, the moment you do that you use lose support for your modified package. So most Red Hat customers probably don't take advantage of the fact that all Red Hat software is "open source", at least not in a direct way. And in that case, why bother trying to use it as a marketing term?
The reason is very simple: Open Source has become a standard way for the industry to produce software, with even companies like Microsoft and Apple participating. So mentioning it in you marketing materials is a bit like emphasizing that your software runs on computers.
I would really _like_ to believe that this is true, however I have never found this in practice. MS and Apple only ever participate in open source software _begrudgingly_. It might be true in some specific sectors (web frameworks, databases maybe), but I don't think it applies across the board, especially not at the business level.
I find it ironic how Open Source was originally coined to pitch the practical benefits of free [as in speech] software to business-types without having confusion between free-as-in-freedom and free-as-in-price and now Open Source has become so synonymous with free-as-in-price that it has essentially the same problem it was coined to solve.
Good note! I've (for quite some time now) just given up on explaining the real meaning of open source. Hey, Stallman is crazy enough to try and fail year after year, let him keep doing that.
The main question missed in the article, IMHO, is: Is this bad for Open Source at all? In the end no. Hey, Open Source is huge! It is bigger than it was and probably it will be even bigger in the future. In the end, I think "Open Source" is just a really bad choice of words to use in a comercial strategy.
I don't think it's entirely fair to compare "then-and-now" screenshots for number of mentions of the phrase "open source". It seems to me that the redesigns simplify, and remove redundant text, which could also be a cause of the fewer mentions.
If however, you're prepared to contract out support and development to a company like Red Hat, then from your perspective as a customer, it may not matter as much. As long as you are willing to pay $$$ to Red Hat and the software/services/support is of sufficient quality for your purposes, you're happy. Now replace "Red Hat" with "Oracle" in the above statement; does it still work? Why, yes. So you really won't care if it's Open Source or not.
At one point there was an argument that Open Source is good because you're free to change to a different vendor. i.e., if you don't like Red Hat's services, you could always switch to SuSE. In practice, alas, the ways in which various boot and configuration scripts are set up are just a tiny bit different enough that it's often not trivial to switch from one Linux provider to another. So the main difference is whether or not you're willing to make changes to the source code and recompile. But if you're a Red Hat customer, the moment you do that you use lose support for your modified package. So most Red Hat customers probably don't take advantage of the fact that all Red Hat software is "open source", at least not in a direct way. And in that case, why bother trying to use it as a marketing term?