Adding a second hatch to the attic
You can use the existing door as an example.
A few things to pay attention to:

Your hatch must be special kind that is air tight when closed. If it's insulated (styrofoam) it's a bonus. You don't need an external hatch door. The airtight fit ensures that no warm and moist air from the house mixes with cold air in the attic to form condensation and related mold problems at the hatch. That's also what the purpose of the existing vapour barrier is, the plastic foil you mention.
The existing vapour barrier must form a closed seal with the door's trim: it is glued or taped or compressed with a foam lip. Your hardware store can show you which sealing method is best for the hatch you chose. The tape and/or glue are special for vapour seals, so make sure you get the right kind.

To cut the opening you need a drywall saw. No need to drill corners first. The saw has a pointy tip to help initial protrusion. They are cheap manual things and easy to use but messy.
You will use your hatch as the cutting template. Your hatch will likely be fitted from the attic side, protruding down, with the vapour barrier lip pressing against the vapour barrier foil and drywall. Then a trim is applied to sandwich the drywall and rim, and provide a finished look. The trim also conceals the drywall edge at your cut.
For best strength, you may need to build a small a wooden frame to the back of the ceiling drywall. This relieves mechanical stresses from repeated opening and closing and prevents cracking of the drywall. The frame is then attached to both joists. One side of the hatch, usually the hinge side, is screwed into one of the joists. The other 3 sides are screwed into the trim or rim of the hatch. For the frame you'll need 2x2 or 2x3 lumber. You will also screw the ceiling drywall into the frame you built. No need to mud the screw heads if the hatch has a trim that conceals it. So make sure that the frame is positioned tightly behind the trim.
If the door is smaller than 15in you do not need to move the electrical wire. However, make sure that while you are working on the hatch and frame that you do not squeeze/pinch the wire or screw/nail into it. You could move the wire a few inches further up the joist.