Option #1:
Use 2 inch polystyrene foam insulation directly against the exterior
walls and tuct tape it to seal it. Erect my 2x4 stud walls against this
and drywall. Also, I could potentially insulate between the 2x4 walls
with regular bat insulation to increase my R value.

Mike Holmes did it this way. he even did the floor the same way.

Option #2;
The old way, as defined by a contractor I spoke with. Lay out tar paper
against the exterior walls. Erect my 2x4 stud walls against this. Fill
the voids with bat insulation and vapour barrier the wall. Put up the
drywall.

Is the basement wall concrete or frame? I'm guessing concrete. I' ve never
heard of using tarpaper on an interior wall. Given the two choices, I'd
pick #1