The rules-as-written interpretation of the rules text is just one way to interpret it, a very particular way to interpret it. See this Q&A for more on precisely what rules-as-written implies.
In particular, there is no expectation that the rules, as written, will necessarily be balanced, match people’s narrative or simulationist expectations, or even make sense. They’re supposed to be, but mistakes get made—frequently, it turns out, with more complex rules systems. The goal of RAW is to be as “interpretation-free” as possible, but as you note, that’s not actually possible—so then the goal is to remove as many subjective factors as possible from the interpretation.
Other interpretations do not have any particular goal of eliminating subjectivity or maximizing objectivity, at least not if it comes at the cost of balance, narrative or simulationist expectations, and/or the interpreter’s idea of what “makes sense.” In most cases, the goal of reading and interpreting the rules isn’t to achieve some quasi-philosophical, quasi-legalistic “objective knowledge” of what the rules say—which is what RAW is usually getting at. In most cases, the goal in interpreting the rules is to maximize the fun had by the table. RAW makes no attempt to do that.
The primary purpose of RAW is to facilitate discussing the rules, particularly without any arbiter (e.g. a game master) present to decide ambiguous situations. That is a useful and valuable purpose, in my opinion, but it is not going to automatically also be the best approach for achieving the best gameplay. “Rules as interpreted” then would be an interpretation of the rules with an eye towards maximizing the quality of gameplay. Which is necessarily a subjective thing, since everyone has different tastes and has different opinions on what the best gameplay is going to look like.
Also note that “rules as interpreted” isn’t a widely-used term, and primarily exists as an attempt to re-define the problematic “RAI” abbreviation to something closer to the way it’s often used—because despite using “RAI” to mean “rules as intended,” many people make no attempt whatsoever to actually establish what the authors’ actual intent was. Instead they just label their own interpretation as “RAI,” which is really problematic for communication (and is on occasion quite rude, to boot). So people tried to “salvage” the term with a “more accurate” definition—but it never really caught on. Instead, it’s widely recommended that you just not use “RAI” in the first place. It rarely, if ever, adds anything to a conversation.
Using “RAI” as meaning “rules as interpreted” can also be problematic when you actually are talking about intent—as sometimes does happen, for example when you have developer commentary outside the rules text. The Sage Advice column that you quote is an example of this, as are the “rules as tweeted.”
For an example of the difference between rules as written and “rules as interpreted,” see this answer. There, I establish what strict-RAW says—that hunger of Hadar applies permanent blindness to every creature ever fully within its area—but then go on to discuss the “interpretation” that the blindness should only last as long as they remain fully within that area. Strictly speaking, it’s permanent, but since that yields an overpowered spell, reconsidering it from the lens that it is actually temporary—which would be balanced—is worthwhile.
This does lead to a semantic argument about what is “an interpretation” and what is “a houserule.” I would defend the above case as “an interpretation” because “creatures fully within the area are blinded” is ambiguous in English, and spells are allowed (and expected) to make exceptions to the general rules. Others will defend far more divergent “interpretations” as being such, rather than a “houserule,” and I may well disagree, even strongly, with that usage. But unfortunately, no one can authoritatively and definitively delineate the distinction, so all I can do—all anyone can do—is make a case for defending their usage of either term. And while I would like for the terms to be clearer—and, ya know, for everyone to automatically use my definitions for them ;) —there is no realistic way for that to happen. It makes communication harder—and sometimes people get defensive about it, as if a “houserule” would be lesser, which I wish they wouldn’t since a houserule is a fine thing—but that’s just how it is. Communication can be hard, particularly if you’re trying to make fine distinctions. Putting care into your language and engaging with your audience is the best—only?—solution. No one gets to dictate terms’ definitions, and spending too much time and effort arguing about them is rarely, if ever, worthwhile.
