Free text is the traditional definition of unstructured data (along with audio & video equivalents). But I agree with GP, that people usually mean inconsistently-structured data when they use the term, whether it's accurate or not.
Free text being the definition of unstructured data is kind of funny if you think about it (and I agree that amongst computer people that's how they think about it), since the whole point of text is to structure data using alphabets, syllabary or logograms, such that it can be meaningfully be transferred and understood by multiple biological interpreters :)
I think what most people understand by "unstructured data" is that it's hard/impossible to define a common schema/template that all instances of the type of data in question would adhere to.
Those are both examples of signals we can't interpret interfering with the signal we're trying to interpret.
However, it's not like the electrical noise came to be by magic -- it's the result of many interactions that have a structure, and hence impart that structure on the "noise", we just lack key facts to be able to interpret that signal.
I think you have this backwards. All data is "unstructured", until you choose to find some structure in it. Deciding a bunch of points fit a line is entirely your call, after all. Until you are there to declare there's a structure, it's just a bunch of dots.
Even having a designated x- and y- axis is some amount of structure, and you're right, a person has to be there to determine how they want to view/interact/interpret the structure. But it would be foolish to say that a csv with "time" and "value" columns is unstructured before a person comes along to plot time on the x-axis.