So what you are saying is, if you think that the take-off edge is what, by definition, determines the name of the jump, then you just don't know anything about skating.
Not at all. My point was that if you think that the takeoff edge is the only thing that determines the name of the jump, then either you don't know enough about skating, or more likely you just weren't precise enough in articulating the definition.
Since this comes up most of the time with regards to lutzes (witness the title of this thread), let's look at a triple lutz as an example.
The definition of triple lutz is not "takes off from a back outside edge, period." There are many kinds of jumps that take off from back outside edge. That definition is incomplete.
The definition of a triple lutz (and feel free to edit for greater elegance) would be something like
"takes off from a back outside edge traveling in one direction, with a toe assist from the toepick of the other foot, changes rotational direction at the instant of takeoff, and rotates three times in the air in the direction opposite to the direction of the takeoff edge."
A perfect triple lutz will meet all of those points of the definition.
There are several ways in which an attempt at a triple lutz could be imperfect.
There could be a problem on the landing alone, including a significant problem such as a fall. That doesn't change the name of the jump, but it does effect how much the jump is worth, in ways that are fairly specifically spelled out under IJS (and short program deductions under 6.0).
But the definition I just offered above doesn't say anything about the landing, so errors on the landing don't have an effect on the definition.
Then there are errors, of several kinds, that represent failures in the takeoff and rotation details that do contribute to the definition. My question is, how bad do those errors have to be before they change the definition of the jump? Do smaller errors of those kinds need to change the name of the jump, or can they just affect the point value?
One way that a triple lutz attempt could fail to be a successful triple lutz would be by rotating less than three times in the air.
If a skater plans to do a triple lutz, which ideally rotates slightly more than exactly three times, but instead rotates slightly more than exactly two times, it's not a triple lutz anymore, it's a double lutz, right?
OK, well, what if rotates exactly 2.5 times in the air? or 2.7? Somewhere in between a triple lutz and a double lutz. What do we call it?
We know what the IJS rules call it. If it's 2.5 revolutions or less, but clearly more than 2, this year it'll be called 3Lz<<. Downgraded triple lutz.
In 2003-04, during the first year of testing the new judging system, it would have been called 2Lz. In casual conversation, under any judging system, a commentator or other observer might say "That was barely more than a double" or "that was an overrotated double."
For purposes of the Zayak rule, it does matter whether the jump is considered a double or a triple. So judges under 6.0 or technical specialists under IJS need to be able to draw the line somewhere between attempted triple and double.
If the rotation is only 2.7 rotations in the air, under IJS this year it will be called 3Lz<, underrotated triple lutz, and will earn 70% of the base value. In casual discussion, especially if the jump was landed on one foot, people might refer to it as "a cheated triple lutz."
If the rotation is 2.8 rotations in the air, it will be called 3Lz and can get full credit, or -GOE from judges who penalize the slight underrotation. In casual discussion it might be called just "triple lutz" or it might be called "cheated triple lutz" or "slightly cheated triple lutz."
There's a range of underrotation error and a range of penalty. There's also a range of names by which triple lutz attempts with various degrees of underrotation might be called. But they all fill the same slot in terms of counting repeated jumps.
Then there are less common errors of toepick technique by which the skater does not take off from a back outside edge witha toe assist from the toepick of the other foot. Maybe the skater accidentally put too much of the of the "picking" foot on the ice, not the toepick but the lower part of the blade. Let's say she manages to get three revolutions and land cleanly anyway. Is it still a triple lutz? For purposes of calling the jump, assigning a base mark, and counting it against repetitions, the answer will probably be yes: it will be called as 3Lz and the judges will reduce the GOE if they see the error.
But if the last thing to leave the ice (for a counterclockwise jumper) was not the left back outside edge but instead the brief right back outside edge of the picking foot, in casual conversation an observer might say "That wasn't really a triple lutz -- she didn't pick correctly. It was more of a triple loop."
Or suppose the skater starts on a left back outside edge, puts the toe of the right foot in the ice, changes edge to a left back inside, and then jumps, taking the weight off the picking foot before the gliding foot, so that the last thing to leave the ice is the LBI edge.
I've done that on single lutz attempts. And more than one coach has said to me "That's not really a lutz. It's more like a salchow." In discussing it, we could jokingly call it a salchow, or a "slutz," or "a flutz with a really weak pick." In an IJS competition it would be called as 1Lz e and I would expect -3 GOE. There's a reason there's no lutz attempt in my programs.

But the judges wouldn't consider it a salchow or even a flip -- it's a really bad lutz.
You won't see one that bad at the elite level, with triples, because it would be really hard to generate anywhere close to three revolutions with that kind of technique. Maybe the worst attempts from Sarah Hughes would come close.
In those examples, parts of the full definition of the lutz jump were not met by the flawed attempts. The ways in which the jumps were flawed made the takeoff or amount of rotation similar in some ways to a jump with a different name. But there's no expectation that as soon as the tiniest bit of error comes in that has the tiniest bit in common with another jump, that totally cancels out everything else about the jump that does meet the definition.
So again I ask, how bad does the error have to be before we stop calling the jump as a flawed attempt at the intended jump and start renaming it to the different jump that it has something in common with.
The renaming can happen on a colloquial level well before it becomes official if everyone knows that there's a degree of joking involved.