Have you ever been a judge in an amateur chili contest? You would not believe what people put in there. I remember one where the guy used bottled spaghetti sauce, hot dogs and lima beans. Granted, he was a US Marine, but still. . .
Those are part of the other Trinity. Mirepoix.
Well, yeah, I know that. But that’s all like Frenchie and stuff, and has nothing to do with chili. 
(And I don’t consider that “the trinity.” Seems like every cuisine has a trinity of ingredients that is important, but to me the term “trinity” is specific to onions, bell peppers, and celery.)
Missed edit: Although I see I said “a trinity” in my original post. Still, “trinity” to me is tightly linked with Creole/Cajun cooking. Otherwise, I’d’ve said “mirepoix.”
Personally, when I make chili, I generally use canned beans and tomatoes, and I don’t find the need to add any liquid beyond the content of the cans. Well, and a splash of cider vinegar at the end, largely as a means of carrying corn starch (though the vinegar itself adds a nice note to the flavor).
I also don’t generally add powdered peppers, since I prefer to use fresh ones. If I weren’t getting enough heat from the fresh peppers, I’d use hotter ones, or more of them. And if I’m cooking for a heat-intolerant crowd, I’d still be adding fresh peppers, and just make most of them bells, as I consider peppers an essential ingredient. I find baffling the notion that chili should contain no solid veggies. Texans don’t like real chili; what Texans call chili is just wet steak.
I would hope so. It’s called “chili” for a reason. 
(After the chile peppers, and bell peppers are a type of chile pepper. Can’t have “chili” without the peppers.) I’ll leave the “real chili” and “wet steak” silliness to someone else.
Of course, some might argue that dried-and-powdered peppers alone, i.e. chile powder or chili powder, would do. I’m sure I’ve seen recipes.
“Chili powder” is usually a mixture of spices, at least in American parlance. “Chile powder” is sometimes used to distinguish itself from “chili” powder, but can be confusing (and, when spoken, is not pronounced any differently than “chili powder” and thus can’t be orally distinguished, unless the chili is specified as in “Ancho chile powder” or similar.) That’s why, as I mentioned before, I tend to like the phrase “powdered chiles” to remove this ambiguity. The Wikipedia article on this topic is particularly bad, saying the mixture is called “chili powder blend,” a term I have never heard used culinarily. In American cookbooks, “chili powder” almost always means the blend. (I say “almost” just to be safe, although I have never seen it mean anything other than that.)
I liken it to curry powder: a convenient blend of spices for the amateur cook.
I never really say or write “chile powder”, unless it’s preceeded by the specific chile name. “Ancho chile powder”, or “Guajillo chile powder”. That helps a little.
The nomenclature gets especially confusing with readers across the pond, as in the UK (and I presume most other English speaking countries) “chilli powder” is the term used for “powdered chiles” or “hot red pepper powder” or whatever you want to call it. So any non-USAian who finds an American recipe containing an ingredient called “chili powder” is going to be missing a few key components (usually, cumin and garlic, and often also onion and Mexican or regular oregano. Some powders even have a few aromatic spices like cinnamon or cloves thrown in, but typically, to Americans, the flavor of chili powder is largely cayenne + cumin + garlic.)
So, to summarize:
Chili powder (US) = chilli powder + cumin + garlic powder + (oregano) (UK)
Chilli powder (UK) = hot pepper powder/powdered chili peppers (US)
I always found the chili restrictions and classifications a little restrictive. I understand that for a “good” chili you want to to have as little “filler” as possible and to let the meat’s flavor shine through. However, there is also a place for flavor variety, texture variety, and also chili that doesn’t cost $200+ bucks per batch.
Also, like Shakes, I’ve always thought the “Texas Red” qualification meant just no beans, and that no tomatoes/onions were just optional to the designation.
As far as the OP regarding thickeners, it sounds like you started off listing all the suggestions that people have given. My only suggestion now would be to start off with less broth/beer and really push the limits of your reduction. That and to dry your veggies before tossing them in. Good luck! I love chili.
Texan here from San Antonio, and the chili made here are tomato based. Some even use kidney beans…gasp!
I don’t think they’re restrictive, but rather descriptive. I have a better idea of what kind of chili I’m going to get, so if I’m in the mood for a beanless, hearty chili, I can order a “Texas red” or “Texas style chili” and be reasonably assured that I won’t get a bowl of beans with some incidental meat. Or, if I see “Cincinnati-style chili” on the menu, I know to expect a soupy, fragrantly seasoned chili (also beanless), but with a fairly unique flavor and texture. If I’m in Chicago and I just see “chili” with no description on the menu, I’m going to assume it’s Midwestern anything goes kind of chili, and not a bowl of Texas red, or New Mexican green, or Cincinnati, etc.
I wouldn’t say that’s necessary for “good chili” in general, just for one specific category of chili. Just what kind of meat you using? Something as cheap as coarse ground or cubed/diced chuck will do, and that’s about $2-3/lb.
“No beans” (except possibly on the side) is the only absolute assumption I make when I hear “Texas red” or “Texas-style chili.” The tomatoes and onions depends on who you’re talking to.
I like all types of chili, even the so-called “white chilis.” In fact, I made one last week with turkey, green chiles, white beans, and even chard. (I have a friend who does a rather good one with fennel bulbs.) But being called “white chili” I know it’s a distant cousin of the Ur-chili that is made with beef (or possibly pork) & chile peppers. It’s not really what I would call a “chili,” but I’m fine as long as it is described as “white chili” to separate it from standard types of chili.
I mean, I consider actual peppers essential, as opposed to just powder made from peppers. That view is apparently not universal, given that there are contests which disqualify any solids other than beef.
I also consider it essential that it at least rate some nonzero Scoville value, so I won’t use entirely bell peppers for any audience. But I can do mostly bell.
And I’ve never tried “white chili” (I presume no tomatoes and an Alfredo-like cream base?), so I can’t say how I like it. I do like Cincinnati chili, though it really isn’t the same sort of thing as no-adjective chili at all.
Yeah, I’m fine with just the powdered chile peppers. To me, they’re more important than the fresh ones, but that’s just me. You can make a perfectly good chili with just dried peppers (and that’s actually the way I tend to do mine.) There’s different flavors between the two. It’s like the difference between paprika and red peppers. Goulash doesn’t taste like “goulash” unless it’s made with paprika (dried red peppers), IMHO. There’s a certain earthiness those peppers impart.
No, no cream. Poultry, white beans, chicken broth, green chiles, garlic, onions, and cumin tend to be the base. You can also add oregano, pepper, some other light colored vegetables, etc., to it. No milk/cream or anything like that.
It’s because the peppers are roasted, which often significantly changes the flavor of vegetables (it’s the only way I can eat asparagus). As you know, some paprika is smoked, and of course chipotle peppers are just jalapenos that have been roasted and smoked.
Well, I suppose it depends on what you mean by “roasted.” Typically, they’re just dehydrated (dried) and ground, sometimes just by being left strung up in the sun. But, yes, that dehydrating process concentrates flavors. I don’t think of it as “roasted” in the sense that they are taken to a temperature hot enough to cause the sugars to caramelize or anything like that, though.
Oh, I absolutely agree that dried paprika is essential to goulash, but then, I don’t consider goulash and chili to be particularly related, beyond the basic level that they’re both stews (this was after stew, but then, so is everything).
And I’ve tried using smoked chipotles in place of fresh jalapenos in chili, and found that I prefer the fresh ones (though I’ll still count the chipotle version as “real chili”, just a real chili that I happen to not like as much).