I'm still standing by what I learned in school. If I steal someone else's work, it is plagiarism. If I create the exact same wording as someone else without their work to go from, it is coincidence.
The online plagiarism checkers came about primarily b/c it was impossible for teachers and reviewers to read the thousands of other theses that had been written previously and determine if you happened to steal your text from them.
As I understand it, they were NOT primarily designed for author's to check whether someone had previously written the same thing as they did. Why should they (author's) care?
Also, there are subtle nuances here - what we are saying for copyright law would most likely NOT apply to patent law or trademark law, although there are similarities.
That said, while I previously would not have - as a result of this discussion - if I intended to copyright something that I wrote independently, I probably would run it through a plagiarism detector and add a statement to the copyright "This work has many similarities to the work of <author name>, but was developed independently of his content." - just to prevent my being accused of plagiarism, which I still don't feel I would have
committed.
There are somewhat musical parallels as well. Vanilla Ice was sued and lost (I believe) b/c "Ice Ice Baby" used the same bass line as Queen/David Bowie's "Under Pressure". If Vanilla Ice knew the bass line and used it without attribution, permission, and royalties, that was legitimate. If he just happened to come up with the same bass riff that Queen used, that is somewhat unfortunate and unfair. If he heard "Under Pressure" ten years prior, but didn't realize he was thinking about it when he wrote "Ice Ice Baby." then ...
Like most things, at least in America, to prove it, the original author would have to file copywrite infringement and sue and a judge would have to decide. AFAIK - IANAL.