indeed, that might be confusing.
However, for independent systems (and thus for matroids) M2 always holds.
In this case F ≠ ∅ and ∅ ∈ F are equivalent.
As you pointed out above: ∅ ∈ F implies F ≠ ∅
Moreover, if F ≠ ∅ it follows that some X is in F and by (M2) all of its subsets. Since ∅ ⊆ X it follows that ∅ ∈ F .
Hence if M2 is satisfied then ∅ ∈ F if and only if F ≠ ∅
However, for independent systems (and thus for matroids) M2 always holds.
In this case F ≠ ∅ and ∅ ∈ F are equivalent.
As you pointed out above: ∅ ∈ F implies F ≠ ∅
Moreover, if F ≠ ∅ it follows that some X is in F and by (M2) all of its subsets. Since ∅ ⊆ X it follows that ∅ ∈ F .
Hence if M2 is satisfied then ∅ ∈ F if and only if F ≠ ∅