(I'm going to try not to get too pedantic here. Would love folks' thoughts of course. But I'm a philosopher talking about philosophy, so it may well be a losing battle...)
Barkman defines philosophy as thinking seriously about something. I think I get what he's driving at, but he also takes it too far very, very quickly. I'm too lazy to dig out my copy so paraphrases will have to suffice, unfortunately. He then goes on to say that Tolkien Lewis are philosophers of mythology because in addition to writing both did quite a lot of serious thinking on what counted as myth, what made or good myth, all of that. Focusing on JRRT (because my knowledge of Lewis is really scandalously slim!), I'll grant you that in "On Fairy Tales" or some of his letters he probably is doing philosophy. He's trying to define and discuss it and give other people good reasons to go along with what he's saying. But speaking as a writer, it's really very hard to write well without thinking seriously about what you're writing. JRRT's trying to create (or re-create, or sub-create) a British mythology. That's part of good world-building. But certainly philosophy means something more than that? Seems too broad to be useful, and pretty insulting to people who don't think of themselves as doing philosophy to boot. (As if a musician or a particle physicist wasn't also thinking deeply about his art/science!)
So let me be a good philosopher and try to do better. As a first step: philosophy means doing serious thought about some subject - philosophically. Which is really horribly circular because it doesn't get us any closer unless we already know what philosophy is. Adding -ly does worse than nothing here. But what it gets at, I hope, is that philosophers don't have the market cornered on serious thought. We're one type of serious thought, done through a certain method.
Next natural question: what does it mean to think (or do anything) philosophically? I'm not going to try to define something humans have been doing for 2,500+ years now, but I'd say most philosophy includes:
1) Identifying a belief hat philosophers would describe as epistemologically virtuous - true, correct, the kind of thing we ought to believe/hope for/have faith in/etc. That kind of thing.
2) Working out in detail why it is good (etc.) to believe (etc.) this thing to be true. Typically this is through writing it out, but I suppose it could take other forms. Important thing is, you're working out the kind o things that should convice any unbiased person - reasons, arguments, etc.
3) Sharing #2 with other people in a way that allows dialogue. Typically this means trying to convince someone you're right, and actually taking it seriously when they explain why they aren't convinced.
Let's put this in fandom terms. Anyone can enjoy Johnlock fanfic. The writer has to think seriously about what that would look like, but the reader doesn't actually have to believe Doyle Mofftiss etc. meant for John and Sherlock to be getting it on - someone who doesn't buy into that can still enjoy a well written Johnlock fic. (I read Sherlolly the same way - not my read of canon but I can certainly enjoy it when done well.) Writing Johnlock meta is a big step closer: here you're explaining why you believe the Sherlock plot will end with John and Sherlock as a romantic couple, or why it should, or why it's good to hope for that even if it will never happen, something along those lines. There's a definite belief, and you're trying to explain why open-minded people should agree.
But that's really just apologetics, making the case for your belief. Philosophy really starts when another fan who thinks Sherlock is asexual and John happily married to Mary comes along, you talk and actually hear each other, and you come to some kind of an agreement (maybe a happy middle, maybe one convincing the other to change her mind). (Worth saying: sometimes it's a long game. I've got friends who've explained why they disagreed with some meta and I've not always responded, we've not always reached quick and tidy agreement but I've mulled over what they said and often over my months at least some of their points make it in to my own beliefs. That's philosophy, too.)
The point is: philosophy is one part making your case, one part actually dialogueing with someone and reacting to that in a productive way. There's a method here. It doesn't look like the scientific method, or the typical creative cycle, but it's a method nonetheless. When JRRT talks about what myths are and actually hopes to convince people, or better yet, perfect his ideas through their feedback, that's philosophy. But when he's actually myth-making, he's doing something else entirely.
Back to the other thing that prompted this: the Times' piece. The author points out that philosophy is very white, and he thinks if we're not going to correct that the least we can do is rebrand it as American and European Philosophy. And... well, I have quibbles with the name, and with the idea that Plato and Aristotle (let alone Augustine) would be counted as Europeans in their day, let alone that American philosophy is at all respectable in terms of philosophy's history, but the basic idea? I can get behind it really easily. Philosophy is hostile to folks who don't come with a certain experience set most common in white men. And the curriculum is definitely old and melanin-deprived.
Thing is, the "white" thing is a bit misleading as I said. The real issue is privilege and power. (Europe, nevermind America, is not synonymous with these things.) And the bigger problem: or a variety of historical reasons, I don't see a lot of non-western voices taking part in the same dialogue philosopher are talking about. Sometimes we don't have enough surviving source materials. Sometimes we don't have the training to parse them like they were meant (Lao-Tzu seems more poetry than nonfiction to this westerner, though I'm not sure that distinction translates all that well). And part of it is we westerners talk about "Indian philosophy" the way we never would "German philosophy" - it's almost anthropology, not really digging into the specific writings of specific figures so much as a broad discussion of that culture's beliefs and values and approach to life, as if anything is that neat and unified.
But the biggest problem for me is Lao-Tzu wasn't in dialogue with Socrates, either directly or through the work of later minds. Sure, we can force them into conversation, but that always seems a bit reductive to me - you're either taking Lao-Tzu somewhere he never intended to go, or else you're only focusing on the part that's relevant to what westerners have been interested in. Not saying non-western philosophy can't be enlightening (look at the free will debate and the way a lot of people use what they call Indian philosophy to reframe the problem of determinism) - but it seems more than a bit racist to say non-western philosophy is only interesting because it solves western philosophy's problems.
Yet I don't have a better answer. In practice, (western) philosophy is where the spotlight is.
Anywho. It's 1:43. I apparently still have some passion and Thoughts (TM) here, but I also have another ghastly workday in front of me. Meaning the path of wisdom probably leads to my pillow just now.