Offering a Draw

I recently played an online chess game where it was my move, and I offered a draw. However, before my opponent accepted, I made a move. Then after my move, my opponent accepted the draw. Is this legal? Doesn't he have to accept the draw before I make my next move? 

No. When you offer a draw on your turn, the offer stands until your opponent makes their move. 

I always thought the draw offer had to be made after making a move.  So the system behaves the same way whether I offer a draw before or after making my move?

you can offer a draw before you move, but over the board your opponent can tell you to make a move before they have to decide to accept or not

When playing live chess, if I offer a draw when it's my turn, and then make a move, the draw offer vanishes. If this is the way it works for live chess, why doesn't it also work this way for turn-based chess?