Ideally, the definition of "opinion based" should be applied very literally. Questions whose answer is objectively true should stay open. Questions whose answer depends on who is doing the answering should be closed by this criterion. Example questions:
- "Does sugar dissolve in water?" - this is an objective question. If one person says "yes" and the other "no", one of them must be wrong. It's a question about objective reality, and a single true answer can be found through empirical observations.
- "What is the tastiest topping for pizza?" - this is opinion-based. If person A says "ham" and the other "mushrooms", nobody is wrong. Each possible answer is correct for the person answering it, but is not universally valid.
Opinion-based questions have proven bad for the network because they tend to lead to one of the following outcomes:
- flame wars. Imagine a question like "are German knives better than Japanese knives". This tends to generate few answers, but heated arguments and lots of upvotes on both sides, plus some vitriol, endless comment discussions, and if somebody takes the time to read the info produced, they are none the wiser.
- big lists. The "tastiest topping" question would likely create that style - at some point, we would likely have 2-3 pages of answers, each mentioning a different topping. The OP might have wanted somebody to help them make a decision, but the result turns out to be a bewildering laundry list of all possible options, some of them posted multiple times across different answers, and the upvote pattern favoring oldest answers.
- answers that are confidently written, highly convincing, and wrong. This is usually when a question is asked in an area governed by localized cultural preferences. For example, look around our questions about cooking rice - pretty much every time somebody mentiones cooking rice with the "add lots of water and drain later" method, the post garners comments saying "you're doing it wrong, you should never add that much water" from people who have never used anything but the "steam with a little water" method. In reality, both methods work well, it just happens that our predominantly US audience has little knowledge of the draining one.
Now, we don't live in a black-and-white world, and objectivity comes on a spectrum. Then it becomes a matter of judgement what to do with questions that are somewhere in the middle. The best criterion is to anticipate when a question is likely to generate unwanted answers like the above. If not, it's left as-is. If yes, we try to come up with a closely related wording which would not produce any problems, and edit the question. If that fails, we close. I see this as complementary to Cascabel's answer: generally, we aim to close "bad subjective" questions and leave open "good subjective" questions. Do we reach that goal? The answer is also opinion-based \-.-/
Now to the two questions you give in the example. I would say that both fall more in the middle of the spectrum than on one end. And yet one was closed and one left open. Why?
The knives sharpening question has a small set of possible answers. One can basically invest the time to learn to sharpen their knives manually, purchase a tool that flattens the learning curve, or bring their knives to a professional sharpener. The preference for one of the three is indeed subjective. But it's very likely that somebody answering the question will be able to give a neutral comparison between these three and thus assist the OP in their decision making. Also, it's not just a "how should I sharpen my knives" question, but asks answerers to evaluate the solutions on a single criterion, "simplicity". This isn't a perfectly objective criterion, but I think it can be estimated reasonably well.
The pan question has a much larger set of possible answers. Pans come in many sizes, materials and shapes. Their usefulness also overlaps a lot - most pans are very versatile. And it also depends on what cuisine one prefers - if somebody doesn't cook Asian, they won't care if the pan would also double as a wok substitute. So looking for a true "most versatile" pan, which is truly the most versatile choice for every single cook, is likely futile. At the same time, it's unlikely to produce a single (or a few) neutral answer that compares the advantages and disadvantages of each possible choice with each other, since there are so many possibilities.
Also, to be very honest - your own answer was one of the reasons (although not the main one) I decided to vote to close the pans question. It looked like the first step towards producing a big-list style of question. Sometimes it's the answers which reveal how to best deal with the question, because 1) they show how the community is likely to interpret the question, and 2) once somebody has an example in front of them, they are more likely to think of a similar example and answer in the same vein. So, once the first person has recommended a type of pan, it's very likely that the second person will also recommend a type of pan. And the third, then the fifth, the fifteenth... and that's exactly what we don't want to happen.
There is an additional subtlety here. When a question is in the middle of the subjectivity spectrum, the best we can do is to answer to the degree to which something objective can be said, without giving a pseudo-authoritative answers to questions which actually don't have an authoritative answer. So, if the tone of the question indicates that the OP is planning to make the final choice, and is asking for additional information to help that choice, we are more likely to leave it open. In the pans question case, I interpreted the question as the OP wanting us to make a choice and tell them the result. That was another reason to close this specific question.
As a short vignette, we recently had a very illustrative old question bumped up: A proper way to cook Mi Goreng noodles?. I would consider the title of the question to be too opinion-based, and thus closable. For pretty much everybody, their favorite method of cooking noodles is the proper way of cooking noodles! On reading the text, it turns out that the OP is asking about something very well defined: their own method produces soggy noodles, and they want to troubleshoot the sogginess. The question was left open, and 12 years later, it has 11 answers. One of them actually addresses the sogginess; ten are simply somebody's favorite method of cooking Mi goreng noodles.
In retrospect, I think that a much better solution would have been to reword the title, while leaving the question body untouched and the question open. That might have produced better answers about sogginess, and most likely wouldn't have goaded people into posting the unnecessary "my favorite way" answers.