While "back kick" is mostly synonymous with "side kick" in most martial arts, there are styles that do have a back kick that is subtly but very significantly different than a side-kick. This is perhaps what you're picking up on.
In a side kick, you will raise your leg up and inwards, chambering the kicking leg in a position whereby the kicking leg is parallel with the ground, the foot is pointing toes to the side, and the flat of the foot facing your opponent. Your base foot (the one on the ground) will pivot as you're doing this, ending in a position whereby the heel points towards your opponent. At this point, you kick straight out, usually striking with the heel or the outside edge of the foot. That's a side kick.
It's a little like putting your kicking leg on the top of a table and sliding it forward across the table the entire time.
In a back kick, you will first pivot your base foot just like you would do with the side kick. The base foot's heel now faces your opponent. Then you bring your kicking leg upwards in front of you, bending the knee, lifting it just like you would lift to prepare for a front kick. Now look over your shoulder to your opponent who is behind you now. Drop your leg straight back down the same way it came up, but keep your knee bent throughout. And continue moving, using the momentum to swing the leg behind you and upwards towards your opponent. The knee straightens out as the leg extends. At this point, most people bend forward due to the limits of their flexibility. And most people have to take their eyes off of their opponent during the final moment of the kick to avoid being off-balance. The strike is done with the heel and with toes pointing downwards or slightly to the side (on a diagonal). The final position of the knee can be either pointing straight down or slightly to the side (on a diagonal), to match wherever the toes are pointing. That's a back kick.
Imagine you're doing a back kick in a one foot wide hallway, with walls touching you on both sides. There's no room for you to bring your leg to the side as a side kick would do. The back kick just sweeps backwards, never touching the walls.
And this is why it is named the "back" kick. The kick is done to the back of you. Whereas the "side" kick is done to the side of you.
Both kicks can either chamber or not. I described both with a chamber. Both kicks can be done while spinning. I described them as stationary kicks.
Notice the key differences between the two kicks. First, the side kick is to an opponent in front of or to the side of you. The back kick is to an opponent to the side or in back of you. The side kick opens up the hip to expose the groin, whereas the back kick does not. The side kick operates in a plane parallel with the ground, whereas the back kick operates in the plane perpendicular with the ground. The side kick allows you to keep an eye on your opponent at all times, whereas the back kick is momentarily blind at the end of the kick. The back kick also exposes your back to your opponent. The side-kick can be (and usually is taught to be) done with the blade edge of your foot, but the back kick is always done with the heel or with the whole bottom of your foot.
The back kick can look more like a side kick during the final moment of the kick. This is when most people, in order to get the best mechanical advantage, will open the hips up a little (like a side kick) while turning their knee and toes to point diagonally to the side instead of straight down. But the lead-in to the kick will have a closed hip and will simply arch straight down and then back up. It will not cross the plane of motion (ie, it won't move off-center, to the side) at all, unlike the side kick.
There are pluses an minuses to both kicks. Stylistically, Taekwondo tends to teach the side kick and avoids doing the back kick. This is probably due to the preference of being able to see your opponent always. The back kick isn't as safe in that respect. Also, the Taekwondo side facing stance (also known as a "back stance" or a "fighting stance") has you in a position to immediately perform a side kick, whereas you need extra movement and setup to be able to perform a back kick. That makes the side kick the favorite for Taekwondo. Taekwondo also likes to kick to the head, and that's more difficult to do with the back kick while not also compromising balance. Balance is another problem with the back kick in general, relative to the side kick, due to the way you bend forward during the kick.
I see the back kick used most often in karate styles such as Shotokan karate. They will do the back kick with a slight turn of the leg at the very end of the kick so that the knee and toes point diagonal at the end. It also permits them to keep an eye on their opponent more than a "pure" back kick would.
One interesting type of back kick is the "mule kick" from Hapkido. They will tend to bend forward completely during the kick, ending with the head as low as your knee or ankle height. Hands can be placed on the ground to brace the kick and improve balance. This permits them to look behind and upwards. It allows them to kick to the upper chest and head of their opponent. And it allows them to make a very strong kick in a more compressed, short range compared with the side kick. It's often presented as a counter to someone doing a bear-hug from behind. You break free of the bear-hug (maybe by way of a back head butt while pulling his arms apart) and fake falling forward to the ground while kicking backwards at your opponent.
Rarely do I see a back kick being used offensively. It's almost always a defensive-only kick. You do it while your opponent is moving towards you and take him by surprise. The kick is difficult to see and avoid, because it's coming from down low and comes straight out without telegraphing. Whereas the side kick is usually done while advancing towards your opponent, offensively. And there is more telegraphing when you do a side kick than a back kick, due to the chambering and the motion to the side which can be seen by human eyes better than motion down low and diagonal towards the eye.
The counter to both the side kick and the back kick is to simply step inward and to the side of your opponent. This leaves your opponent in a very vulnerable position. If your opponent does this to you while doing a side kick, you can see it while it is happening and try to modify your kick into something else. If you're doing a back kick, it's harder to see what your opponent is doing, and recovery will be harder also. The dive front roll or a leg sweep are things you can practice as an escape from a back kick gone bad. There are a number of kicks that Capoeira practices from this vulnerable, bent-over position which could be helpful to you in recovery.
Yeah, I'm surprised this isn't discussed more.
Hope that helps.