If the requirements to be a true dragon are have the dragon types and have age categories, then it seems like a half-red dragon human would be a true dragon right?
This issue is a mess. Please start with a similar discussion of the dragonwrought feat; many of the same arguments from there apply here.
Monster Manual defines “true dragon” as just one of the Dragon-type creatures listed in the “Dragon, True” entry. Half-dragon isn’t there, so it isn’t one by the Core rules.
Draconomicon claims to overrule Monster Manual on the subject of dragons (cf. the arguments about such practices vis-a-vis Rules Compendium), and provides a definition of a true dragon, specifically being that it “advances through age categories.” What that means is a subject of a great deal of argument, which is covered in greater detail in the above-linked dragonwrought question.
The long and short of it is that, RAW, half-dragons count as true dragons if their non-dragon race has age categories, and don’t if the non-dragon parent doesn’t. Yes, that’s insane. But if their parent does, then they have age categories that they progress through as they get older, and doing so grants them “more abilities” (rather literally, in the form of higher ability scores) and “greater power” (those higher ability scores mean they’re greater at some things than they were before – and as dragons, true or not, they don’t take age penalties). That meets the definition. Draconomicon never says anything about requiring the “draconic” age categories in the definition.
Races of the Dragon: yes, super-strictly
Races of the Dragon once again attempts to assert primacy on the issue of true-dragon-ness. It also includes a section on half-dragons, which states:
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).
Here, Races of the Dragon defines half-dragons as “versions” of the full-blooded true dragons. The entry doesn’t have, for example, half-red-dragon, it has just “red dragon” and then gives stats for the red half-dragon. As such, it states that a red half-dragon is a “version” of a red dragon, and is therefore a true dragon (and, in fact, a red dragon as well).
Intent: impossible to say
The distinction between true dragon and not is only rarely relevant: one must be a true dragon to have a Dragon Psychosis from Dragon vol. 313 or a Sovereign Archetype from Dragons of Eberron. Aside from these, however, the rules never make a distinction between true dragons and other dragons.
The authors of that Dragon article and Dragons of Eberron were not the same as the authors of Monster Manual, Draconomicon, or Races of the Dragon. They may not have even discussed this issue with those authors, or been aware of the various definitions. They also did not address half-dragons or dragonwrought kobolds specifically. So who knows what they expected?
Half-Dragons are not True Dragons.
The known varieties of true dragons (as opposed to other creatures that have the dragon type) fall into two broad categories: chromatic and metallic. The chromatic dragons are black, blue, green, red, and white; they are all evil and extremely fierce. The metallic dragons are brass, bronze, copper, gold, and silver; they are all good, usually noble, and highly respected by the wise.
All true dragons gain more abilities and greater power as they age. (Other creatures that have the dragon type do not.) They range in length from several feet upon hatching to more than 100 feet after attaining the status of great wyrm. The size of a particular dragon varies according to age and variety.
Note that it says "as opposed to other creatures that have the dragon type" and "other creatures that have the dragon type do not".
Since Half-Dragons are not a "Red Dragon" but rather a different kind of creature with a template that is based on a Red Dragon (for example) and happens to have the Dragon type, they are not considered True Dragons.
Yes/No/Maybe (Depends on the books used)
The M&M specifically states a list of true dragons, which does not include the half-dragon template. Therefore, according to the M&M, a half-dragon is not a true dragon.
However, when we use the Draconomicon, it states that a true dragon has age-categories (which makes it a giant mess). The reason for that is that suddenly, we get to a situation where a human half-dragon is a true dragon as it has (human) age categories while a Allosaurus half-dragon does not, as it lacks age categories.
There is further relevant text in the Races of the Dragon but those are already explained in the answer by KRyan.