You sir are correct.
Normally you write a subgroup N is normal in G iff for all . This is your definition given, but you can also notice that so this definition also tells you:
Which is the second conclusion you drew.
The definition of a normal subgroup is: for each element, n, in N and each g in G, the element gng−1 is still in N where N is a subgroup of G. Can this be changed to ...element g−1ng is still in N. I think it still makes sense since gng−1=m for some m in N thus g−1gng−1g=g−1mg which gives n=g−1mg . Now g−1mg is in N since n is in N.
Is this correct?