...At the far end insulator, you will NOT have a connection between the 'extra' wire and the main antenna line, so the 'extra' wire IS a part of the antenna radiating section and must be considered part of the TOTAL antenna length.
It actually is a bit more complex than that: folding an insulated wire back does raise the
resonant frequency, but not by as much as it would if the wire were bare and connected back
to itself. (But even that would actually lower the frequency slightly, as the end section is
thicker with the folded back wire, so has more capacitance.)
This is another problem with using the "traditional formula" that seems to assume that wire
length is constant for different antenna shapes: it isn't.
I built an 80m dipole with about 5' of wire hanging down at the ends, and put an insulated
clip on the end. If I clip it out the rope as far as it reaches (so the wire is fully straight)
I can get down to about 3750. By folding it back up the wire the resonant frequency
moves to about 3960, allowing me to operate all the way up to 4 MHz. I can adjust the
location of the clip for resonance anywhere in between, all with the same length of wire.
(This is convenient for portable operation, as it allows me to make minor adjustments to
the antenna tuning as needed each time I set it up.)
Most of my antennas use stranded, insulated wire, because I get spools of hookup wire
cheap for experimentation. It is also much more practical for portable antennas, as it
doesn't kink nearly as much as solid wire and is easier to wind up and put in my pocket.
With thick wire you can just twist the wire end around the standing part as you would
with solid wire. With hook-up wire I usually tie a loop knot in the end of the wire (a
bowline or a figure-8-followthrough works well.)