Here is an example of a polynomial that passes the 2^n - 1 test but is not primitive.

polynomial: x^4 + x^3 + x^2 + x + 1 (11111)

ELEMENT  ORDER
0001     1
0010     5
0100     5
1000     5
1111     5
0001     1
0010     5
0100     5
1000     5
1111     5
0001     1
0010     5
0100     5
1000     5
1111     5
0001     1