Here is my philosophy on kicking skills, not only at various Taekwondo schools, but throughout the Martial Art in General.
When you attempt to kick an opponent,
your foot should leave the ground and strike the target. 
Simple enough?
Further explanation:
You should try not to fall down when kicking, so
balance is important, However, if you do fall down, kick on the way, or kick from the ground.
Your foot should not miss the target, therfore
accuracy is important.
You should hit the target hard enough to do damage (unless it is a fake or feign), thus
power is important. If the kick does not do the desired damage, then you need to change how you are doing the kick (see an instructor who can demonstrate and/or teach how to do the kick with power).
You must be able to avoid telegraphing, therefore the kick must be
fast, and
desguised, without advanced warning (unless you are intentionally misdirecting the opponent's attention).
You must be within range to reach the target, thus
distancing is important. If it is difficult to close the distance, allow your opponent to make an attack, closing the distance for you.
You must be able to kick when your opponent is not prepared to defend, avoid, block, or grab your kicking leg, therefore
timing is essential.
It might be difficult to land a kick if you initiate the attack, thus it is important to work on
counter-maneuvers.
One kick is seldom sufficient to break through an opponent's defenses, therfore, it is wise to practice
combination kicking using mulitple kicks, and hand techniques in a single drive.
Having multiple target options is good, thus
flexibility and agility are extremely beneficial. Kicking the knee is quick and effective, but is often defended against. Kicking the head can be risky if done improperly, but often results in a knock-out. As well as high kicks, flexibility is for reaching a distant target. Know
when to do which,
and when not to.
Kicking is a powerful tool, and should be practiced to its fullest extent, in order to be able to call upon it when needed. However, kicking is not always feasible, thus you should balance your training with other strikes, holds, throws, and general grappling and self defense skills.
Don't be "limited" in your arsenal of possible weapons and techniques.
Basically, you must be able to kick your opponent, with sufficient power to do damage, and not miss. If there is a Taekwondo Dojang (or any other Martial Art school) that teaches differently, I have not heard of it.
The differences you might encounter are between the percentage of desired use of kicks vs. hands (should be, whatever works at the time). Also, if speed and sneakiness is desired over wide range power (should be, whatever works at the time). Another difference is cocking positions of the kick before you extend it (short and close, or long and wide). This includes the side kick from near the supporting knee, or extended out under the kicking knee. Some people use more straight leg, from the hip kicks (crescent, hook, spin wheel) as opposed to a tight cocked position that uses more of the knee action at the moment of extension.
Some teach different degrees of pivot on the supporting foot to allow further rotation of the hips, while others keep it minimal to be faster, more defensive, and flow with combinations.
You will see a preference in some schools for low kicks, and basics on the ground, while others kick high, kick fast, kick often, jump, spin, and fly through the air. The techniques are usually taught pretty much the same, but the preference for caution over devastation might be the standard school of thought.
While some organizations, and Kwans are going to have traditional methods, and recommended approaches, I find that most of the differences are optional preferences that can be effectively applied in most situations, and the method of training is left up the the instructor. I tend to include instruction in each method, and give students the option of which to use in an actual self defense scenario.
You will find that the names of kicks might vary from one school to the next, but this has little significance outside of the Dojang. You are not likely going to tell an attacker in the street the name of each kick before you kick them with it. Although, after you kick them, many opponents might lay on their backs, seeing stars in front of their eyes, and say, "what hit me?" :btg:
Perhaps other Intsructos here at MT who follow a more strict "Kwan" approach, or more streamlined modern methodology can say which is their preference for kicking variations.
CM D. J. Eisenhart
