General misconception about $\sqrt x$
They probably just think that $\sqrt{x}$ means any number whose square is $x$, and don't know that the definition is just the positive root. I don't really think it's anything more than that.
This is entirely context-dependent. First, despite pretenses in U.S. schools, for example, there are no "rules" in mathematics, and certainly no enforcement mechanisms. Further, although it is undeniably a good thing to encourage "careful thinking", to say that this is identical to "logic" is a misrepresentation, as the latter tends to limit its subject to "what can be entirely formalized", while mathematics itself posits no such constraint.
In particular, although pointless ambiguity is not a plus, attempting to "define/control" usage to remove reasonable ambiguities is (I think) at best misguided. If nothing else, rules that have some sense in one context may fail badly in others.
Thus, while there are certainly reasons to sometimes declare $\sqrt{x}$ to be the unique non-negative real square root of non-negative real $x$, there are certainly contexts in which it'd be convenient to allow it to refer to any real square root. And, of course, when taking square roots of complex numbers, there is an inescapable issue of specifying branches, etc. (No, the phrase "principal square root" doesn't really resolve things, because analytic continuation transgresses the declaration that we "always take the principal branch".)
A more vivid example is the age-old discussion of "whether 1 is or is not a prime". First, well into the 19th century many serious mathematicians did refer to it as a prime. The main disadvantage of doing so is that statements of results tend to be messier. Thus, the linguistic or conceptual advantages of saying 1 is prime are outweighed (as it turns out) by disadvantages, so nowadays we say it is not. Nevertheless, one can easily find on-line arguments purporting to "prove" that it is prime, or "should be".
About square roots, in any circumstance, I absolutely do not trust that whoever's writing will conform to whatever rules they or anyone else might claim to prescribe. I myself certainly have no "rules" about this, but would prefer to emphasize explicitly the single-valued-ness or two-valued-ness or complex-variables-ambiguity as context demands.
In fact, attempting to "resolve" the question on grounds of "rules" or "logic" may obfuscate the very real issues about the fact that there are two square roots, branches with complex numbers, and so on, as though those were somehow illicit.
And, e.g., having answers depend on careful attention to the articles "a" or "the" sounds like a trick question. Also, even if we grant that "the" means "just one", it's not the case that "the" means "the unique positive one, if it exists"... This level of fragile formalism isn't really very useful.