The person in the space ship will experience time twice as slow as people on earth. So the person in the spaceship would expect people on earth to age twice as quickly.

I targeted this part of your reasoning. Time on spaceship is moving slower (in a sense) than time on earth in reference frame where earth is stationary, yes, but it doesn't follow that time on earth therefore moves faster than time on spaceship in reference frame of spaceship, quite opposite.

t'=\gamma(t-vx/c^2)

It is both valid when t is measured in reference frame of spaceship and in reference frame of earth.

Thus time in reference frame of muon is moving slower relative to our reference frame and time in our reference frame is moving slower relative to muon's reference frame.