Merely twisting together any wire is generally considered an unreliable connection, especially when dealing with mains AC power. If this connection will be enclosed in some way that protects it from fingers of the curious or inept, or other kinds of accidental contact, then simply soldering the twisted connection and insulating it with electrical tape (the black kind; your #4 above) may be sufficient. If the exposed copper length is short enough, another option is a wire nut (twist-on pressure connector), which should not require any additional insulation as long as no copper is easily accessible beyond the base of the connector. Most people on these forums would probably prefer to untwist the connection, slide an appropriately-sized section of heat-shrinkable tubing over one of the wires, re-twist them, solder them, then slide the heat shrink over the splice and shrink it. There are other methods that will work safely too.
Option #2, wire nut:
Wire nut steps:
Wire nut properly installed:
Option #3, soldering.
Twist solid-core wire:
Or twist stranded wire, and soldered:
Be sure to crimp and/or trim the ends of the wire tightly against the splice, so that they do not poke through the tape or tubing that you use to insulate the connection.
Heat shrinkable tubing, in place:
Heat shrinkable tubing, shrunk:
Please note that heat shrinkable tubing does not usually shrink much in length, only diameter; the tubing is much shorter in the second image because I got these images from different sites.