Does unique mean "at most one" or "exactly one"? This is a bit of a soft question, but what does or should "unique" mean? Does it mean "at most one" or "exactly one"? I have seen different textbooks use different conventions on this matter.
 A: The difference between "at most one" and "exactly one" is small but important. As it turns out, "exactly one" implies that there is "at most one", but not the other way around. This makes either acceptable as a stand-in for unique, depending on the context.
"at most one x" means that if you have two things, and both of them are x, they must actually be identical. BUT, there is nothing in "at most one" that guarantees there exists an x. Example: "there is at most one prime divisible by 4" is true, even though there is no such prime.
"exactly one x" guarantees that there exists an x and that there is at most one of these x. Thus, it is a stronger statement than "at most one". So, "there is exactly one prime divisible by 4" is false, since there are no such primes.
Importantly, both of these statements capture the idea behind uniqueness. Often, talk about unique objects presupposes that the object in question exists and so "exactly one" works fine.
But, one occasionally gets statements like "unique, if it exists" which correspond to "at most one".
A: It depends how you define "unique".
Uniqueness existential quantification is the concept that there exists one and only one solution or element to satisfy some kind of conditions. The example given on Wiki is that there exists exactly one number n in the natural numbers such that n-2=4; namely, that being n = 6. Thus, it generates a singleton, but not the null set.
This is a more constrained definition of uniqueness than the possibility of there being 0 solutions. In linear algebra, matrices and differential equations often have what is treated as three different types of solutions:

*

*one (1) unique solution

*no solution (0)

*infinitely many

The result is based on how the matrix / differential is constructed and constrained. In the instance in which one has 0 solutions, there could be close approximations, but those are still not direct solutions. The question then becomes whether or not no solutions is a solution or just a statement of fact. By contrast, if the differential isn't constrained enough, then there could be infinitely many solutions, and thus there is no uniqueness in the solution at all.
Limits are also treated in an interesting way in that sometimes the limit simply isn't computable due to asymptotic divergence, etc., and thus we say that the solution is nonexistent or "the limit does not exist" (DNE), meaning that it doesn't converge to a single solution, discrete and finite, but rather goes off into infinity. That is the answer, but do we call it a solution, or a lack thereof?
Further, there's so-named "essentially unique" where objects are unique solutions only due to the fact that all other solutions are identical through equivalence relations. This is the basis for the Fundamental Theorem of Arithmetic.
I'm sure there are more advanced examples in analysis, number theory, combinatorics, and topology that I have no idea what they mean, including things being equivalent up to a certain isomorphism, but I think that these examples are reasonably expressive in the types of uniqueness that can exist in mathematical discourse.
A: *

*

Does unique mean "at most one" or "exactly one"?

The word “unique” per se never just means at most one.
While the entire phrasing “a unique solution, if it exists” means at
most one (i.e., either $0$ or $1$ solution, depending on whether
it exists), the word “unique” itself in this context simply means
exactly one. (After all, having no solution entails having no unique solution.)


*

Does unique mean "at most one" or "exactly one"?

In (points #3 and #4, interpretation $(3)$ of) my self-answer to
“Unique” doesn't have a unique
meaning, I gathered
that mathematicians regularly use “unique” to mean exactly one.
However, if this definition is taken literally, then “every column
has a unique digit” is true of both the second and third rows:
\begin{gather}6,4\quad 7,5\quad 8,0\quad 9,0\quad \\ 6\quad\quad
7\quad\quad 8\quad\quad 9\quad\quad \\ 7\quad\quad 7\quad\quad
7\quad\quad 7\quad\quad \end{gather} In view of this, the second row
is more clearly described as “every column has a distinct digit”
(i.e., each row has exactly one digit, which no other row has),
while the third row directly as “every column has exactly one
digit”.
(To be clear: I'm pointing out that it sounds weird to describe
every column of row three as having a unique digit.)


*For completeness, I added the first row to illustrate a third (and
typical) usage of “unique” in common parlance, one that connotes
neither exactly one nor at most one: here, “every column has
a unique digit” actually means that each row has at least one
digit that no other row has.
