I think n==3 and n>=4 is not fully correct.

when n==3 and m==2, possible states are as below

Initial state : 111

Flip All : 000
Flip even : 101
Flip odd : 010
3k + 1: 011
Combination of 1 and 4: 100
Combination of 2 and 4: 001
Combination of 3 and 4: 110

Similarly,

When n==4 and m==2, possible states are as below

Initial state : 1111

Flip All : 0000
Flip even : 1010
Flip odd : 0101
3k + 1: 0110
Combination of 1 and 4: 1001
Combination of 2 and 4: 0011
Combination of 3 and 4: 1100

In both the above cases, the initial state cant be reached. But, initial state can be reached if m >= 3 and hence m >= 3 has 8 states

example: when n == 4

Initial state: 1111

Flip even: 1010
Flip odd: 0000
Flip All: 1111