I printed out your two link problems and looked at the language. It becomes clear that the word "involution" is used by "grobber" as any instance of a projective transformation that returns to the identity when repeated. This would include reflection in a fixed line, inversion across a fixed circle, rotation of 180 degrees around a fixed point (this one is called reflection in the point, evidently). The other language on pencils of lines and the like makes it clear that we are talking about projective geometry.
The attitude that is somewhat new to me is the statement that "there is one and only one of these mappings that accomplishes ----, further work shows that the particular mapping is __." The more familiar application is in Moebius transformations, where any three distinct given points in $\mathbb C \cup \infty$ can be sent to any other three points, then the particular transformation that does it can be worked out if necessary.
So, I am convinced "involution" is a collective word, as in "Moebius transformations." The Dover titles I mentioned include some that emphasize elementary problems. I can also point out that Coxeter was very fond of projective techniques, and some of his books are available.
EDIT: I found only a few other mentions of the word "involution" on AoPS that clearly relate to projective geometry. The most useful was about the Desargues Involution Theorem. This is shown, in detail and in somewhat outdated language, in section 63,pages 265-273, of a Dover reprint (and translation), Heinrich Dorrie, 100 Great Problems of Elementary Mathematics.
I also found plenty of lesson material online, professors writing their own versions of the theorem. Here is the one that sent me to Dorrie