Well, that's my interpretation. Other interpretations are equally (or more) valid. :-) Basically as far as I can understand, the only thing we know about the Dwarfish language comes from the name builder. As you say, the rules are strange, but I think that half of what's great about DF is the serendipity of the strange rules it has.
A quick grammar lesson (I used to teach English as a foreign language so this is fun for me. If it gets tedious, feel free to say so!) An adjective is a word that modifies a noun. For example "blue" is an adjective. I can say "blue car" and we can understand that the car has the attribute of "blueness" to it. A verb (in this case) is an action. "vault" means to jump over something. If I say "The blue car vaults", you can understand that the car (which has an attribute of blueness) is jumping over something. If we can imagine the car vaulting yesterday we would say, "The blue car vaulted". "vaulted" is known as the "simple past" form of the verb.
In English there are also forms of the verb called "participles". Participles act like nouns, adjectives or adverbs. The "present participle" is formed by adding "ing" to the verb. For example, I can say "Vaulting is fun". In this case "vaulting" is a noun-form of the verb. We realise that the action of jumping over things is fun. I can also say "the vaulting blue car". In this case it acts as an adjective (describes the noun). Not only is the car blue, but it is also vaulting over something (right now!) Finally, we can use it as an adverb (describes the verb)... well theoretically anyway.... um... "the blue car vaultingly drives forward". Well, I don't think you can actually use the present participle as an adjective with "vault" :-) But even this made up sentence is probably understandable to you. The driving is jumping over things.
As well as the "present participle", there is also a "past participle". In English, this usually confusingly looks just like the simple past (English is a crazy language and I pity anyone who has to learn it). The past participle for "vault" is "vaulted". Imagine that I vault over a blue car. The blue car becomes a "vaulted blue car", doesn't it (indicating that I jumped over it sometime in the past). Other forms of the part particple are a bit trickier. I can say "Having vaulted, I was happy". In that case "vaulted" is being used as a noun/state. Now that I am in the state of having vaulted, I am happy (because vaulting is fun). Again, I can't really think of an adverbial form.
Now, English is not just confusing, it's annoying. Sometimes we use the past participle as an adjective so much, that it takes on a new meaning. When I say "I jumped over the car. The vaulted car was small", it's clear that "vaulted" is the past particple of "vault" -- meaning something I have jumped over. But when I say "The vaulted ceilings are very high", I'm definitely *not* saying that I jumped over the ceilings. Instead, there is a completely separate word "vaulted" which is an adjective. It means something that has the shape of an arch. It was originally derived from the verb, because when I jump over things I make the shape of an arch, but it has diverged in meaning.
So when you say, "Vaulted anger", you almost certainly mean the adjective "vaulted" (having the shape of an arch), not the past participle "vaulted" (something I jumped over).
Getting back to Dwarfish: In this case the name generator allows sentence fragments of the form:
<front component><rear component> The <first adjective> <second adjective> <Hyphen Compound>-<X> Of <Y>
Notice that the only spaces in the sentence fragment come in the "The" part. So if you want spaces in the english translation, you must use "The X", "Of Y". Also, I should point out that you can *only* use the adjectives with "The X" (which will turn out to be important later). If you try to set an adjective without setting "The X", it won't be visible.
Each one of these parts has restrictions. The front and rear components allow verbs, particles and nouns. The adjectives allow only adjectives. The hyphen compound allows adjectives and nouns. "The X" allows only nouns. "Of Y" allows only nouns and present participles (-ing words).
Anger is a noun and you want to describe the anger with an adjective (vaulted). There are only 3 places you can put the noun in this case: rear component, "The X" or "Of Y". If you put it in the rear component, you won't get a space in front of it. If you want the space, you can only put it in "The X" or "Of Y".
The more difficult problem is "vaulted". You are using "vaulted" as a adjective, I believe (it has the shape of an arc). "Vaulted" can not be used as an adjective in Dwarfish. It only has the past participle form. So you can only say "vaulted anger" with the meaning "anger that I jumped over".
Having said that, it means that you can only put "vaulted" in the first or second component. If you put it in the first component and put anger in the second component, then you get "vaultedanger". If you put it in the second component, then you must put "anger" in "the X" or "Of Y". "Of Y" makes no sense, so you are left with "vaulted The anger".
Which is better for your purposes? Really, we would rather have used "vaulted" as an adjective if there was an option to do so. If we could do so, what would our options be? Note again that you can *not* put adjectives in the first or second component! You can only put them in the first or second adjective place. And you can only add an adjective with "The X". The name generator will not let you add adjectives without the "The X" portion.
So this means, *if* we could say "vaulted anger" in the sense that you want it (with "vaulted" meaning having an arch shape), the we *must* say "The vaulted anger". Since there is only a past participle form of "vaulted", my personal feeling is that choosing to use the "The X" shape rather than using first and second component, you are pretty much emphasising the adjective role of "vaulted", even though it is a past participle.
In my mind, "vaultedanger" means anger I have jumped over and "vaulted the anger" means anger having the shape of an arch. Of course, this is *not* how it reads in English, but foreign languages often have those kinds of subtleties. I have no idea what Toady was thinking when he built that name generator. Probably not what I'm thinking when I look at it 10 years later :-) But the weird rules have a kind of life of their own. It's part of the fun!