Actually, it's more accurate to say that middle C on the guitar is the fifth fret G string, or 1st fret B string. You're right, when we see a middle C on the page, we play it at either of the locations you mentioned. However, the guitar is a transposing instrument. It plays an octave lower than it reads. If you ask anybody else for middle C, the note they will give you is the one that sounds at the fifth fret G string, or first B string.