Taekwondo exists in every one of the alternative forms you mention, Bob.
Military: Simon O'Neil has a very nice discussion of military TKD in his 9th Combat TKD newsletter. He notes that
The function of Taekwondo in the Korean army of the 1950 was that of killing... Taekwondo's purpose was to enable soldiers to incapacitate and preferably terminate an enemy with their bare hands if a weapon was not available. There are several accounts of Korean military men doing precisely this during the Korean and Vietname Wars... [to achieve this], it was essential that the techniques be learned in as short a time as possible, and that they should offer substantial guarantees of success even under highly stressful conditions. This mean that the core methods should be few and simple
The core techs that SO'N details—techs which are reflected in the Chan Hon tuls, as he notes, and a number of later KKW forms—are power strikes, throat attacks and damage to the head and neck, especially head twists, for the specific purpose of breaking an enemy's neck. As SO'N notes, during this earlier, pre-sport phase, TKD was known in the West for extremely hard power strikes and no-nonsense low/mid kicks whose clear function, as in the ROK army, was to severely damage an attacker's limbs or other body parts with the goal of setting up one of several possible killing finishes. The terrifying use of these techs at the battle of Tra Binh Dong in 1967 is now a part of military history.
Street Defense: The TKD toolkit has many of the same techs that karate does, where the latter was designed primarily for civilian street defense, hence incorporates a lot of moves which appear to be particularly useful in responding to typical assault initiations along the lines discussed by Bill Burgar, citing Patrick McCarthy on `habitual acts of violence'—the kind of thing we're used to seeing as the lead-up to a violent attack. Grab-and-punch is still one of the all-time favorites, and both karate and TKD—which insofar as it's a fighting art, is simply Korean Shotokan—have many of the same technical responses to grabs, e.g., wrist trap/elbow pin followed by strikes to the head with fist, knifehands or, probably most devastatingly, elbows. Unlike the military situation, the techs involved in controlling the attacker would not, in the normal course of things, have been ramped up to the level of a killing (counter)attack, and could be adjusted to allow a finer-grained response than would have been available to a heavily-laden soldier under CQ H2H combat conditions.
Sport: The killing/damaging techs are suppressed, and CQ methods, crucial to survival in military or street fighting, are replaced by longer-range methods, corresponding to the completely artificial conditions of tournament competition (a trained opponent, as vs. untrained assailant, who is trying to score points rather than kill you or damage you, etc.) Kicks, which give the contestant the best chance of scoring points while staying safely out of range, become the paramount weapons, and the scoring system is designed to favor kicks which are both spectacular and technically difficult. Hand techs are virtually elmininated, and high head kicks become the summum bonum of the arena.
Health/fitness: Ironically, this is the role which TKD now apparently plays in the ROK army, according to SO'N. Many people train TKD primarily for exercise; kihon line exercises are an end in themselves, and thing like cardio kickboxing can be usefully seen as the final dilution of the martial content of TKD.
It's clear that the TMAs are sufficiently versatile that any of these variants can be realized. The range of their toolkits, and the available training methods, pretty well guarantee that just about any level of violence, from nil to max, can be applied, along the lines illustrated in in the foregoing. And then it just comes down to what you want...
