There are three somewhat-contradicting definitions of True Dragon, from Monster Manual, Draconomicon, and Races of the Dragon (listed by publication date).
Please also note my other answer, which poses a rebuttal that I’ve come across.
The first definition, in the Monster Manual, is simply the “Dragon, True” entry, which lists ten creatures, five “chromatic” and five “metallic.” Several later books included similar groupings of dragons, such as “gem” dragons and “planar” dragons, as well as individual creatures with similar stats. Some of these creatures explicitly stated that it was a True Dragon, in addition to the ten from Monster Manual, but others did not, leaving the question somewhat ambiguous.
Draconomicon took it upon itself to attempt to define a True Dragon, explicitly superseding Monster Manual on the subject. Unfortunately, because of the haphazard nature of the existing True Dragons (and similar creatures that the authors wanted to include under the banner), the definition relied on an almost incidental quality: the presence of age categories. The True Dragons in Monster Manual and elsewhere had twelve; the other creatures of the Dragon type, like most monsters, did not have any.
The official definition became “advances through age categories,” itself a particularly troublesome phrase due to the verb “advances” and the way True Dragon stat blocks were laid out, namely the way they included all twelve age categories in one block to save space. Thus, many believed that this wording required that a creature have something likened to “Advancement: By age category.”
The problem with this is that the rules do not officially recognize any such thing at all. True Dragons merely have “Advancement: By HD,” which are then broken down into entries for each age category as each is a separate creature.
It is entirely possible that the authors of Draconomicon intended to refer to the organization of True Dragon’s stat blocks as the deciding factor, but it is uncertain, and in any case it’s unclear that they even could, since a typesetting choice like that would not normally have any rules significance.
The alternative reading is that “advances” was merely a particularly poor choice of word, when the synonym “progresses” or similar could have been used without ambiguously maybe-referencing the Advancement section of monsters’ stat blocks. In this reading, a Dragon-type creature merely needs to have age categories to progress (or advance) through in order to be a True Dragon.
It is worth noting that by this definition, the Dragonwrought Kobold is a True Dragon. It does have age categories; it even has the traditional “draconic” age categories (which Draconomicon does not explicitly require).
Races of the Dragon
Races of the Dragon added and updated a lot of material to all things Dragon, and as a result has several sections with the phrase “this material supersedes any from Draconomicon.”
The most relevant such section begins on page 69, in a section dealing with Half-Dragons. The exact statement in Races of the Dragon is as follows:
The half-dragon template presents special attacks and
special qualities for half-dragon versions of the ten varieties
of true dragons described in the Monster Manual. The information
here expands that list to include all true dragons
published in DUNGEONS & DRAGONS products to date. It
supersedes any other previously published information
on this topic (such as from Draconomicon).
Considering that this section is supposed to be about half-dragons, it is odd that it is most relevant, but it is. The most important part of this section is this section “expands [the list of half-dragons] to include all true dragons.” This statement is not qualified anywhere in Races of the Dragon: it is an absolute claim that the list includes all true dragons. As a result, if a creature does not appear on the following list of half-dragons, it is not a true dragon.
Conclusion: all true dragons have a half-dragon version, and lack thereof is, per Races of the Dragon, indication that a creature is not a True Dragon. Dragonwrought Kobolds are not on the list of half-dragons that begins on page 69.
Since Dragonwrought Kobolds are introduced in Races of the Dragon, and no book after Races of the Dragon was ever published that claimed to supersede Races of the Dragon, there is no possible combination of allowed sources that would result in Dragonwrought Kobolds being True Dragons.
Dragonwrought Kobolds, therefore, are not True Dragons.
(sidenote: half-dragons, on the other hand, are. Not only are they True Dragons, they are “versions” of their full-blooded progenitors. In other words, a red half-dragon is not just a True Dragon, it’s also a red dragon, and would qualify as such if any requirement specifically required one be so.)