I think I quite understand what you mean, but I have difficulty in undersatnding what you mean by the third note of the scale.
So if we have A, B, C, D, E, F and G
With A Major we fret to obtain A, C# and E. Shouldnt the third note among these 3 note E?
With E Major we fret strings 3, 4 and 5 obtaining respectively: G#, E and B, but we flatten G# to obtain a G. In this case the third of theses notes is G, so it complies what you say.
Thus, I believe I don't understand what you mean with the "third note", unless you mean the third note to sound when you perform a downstroke. Do you mean this then? Most surely... ;)
Nevertheless, in the case of C an Cminor the note we change is the second... from E to D# and the strings that sound change...
Sorry bor being a nuisance. I just enjoy learning with some detail, but I know I can be a pain in the neck.. :P