Magic the Gathering Question?

I was playing some magic at a local FNM yesterday and after that was over, my friend and I decided to have a few friendly games. I decided that I wanted to play my "Liliana's Specter" and in response to that, my friend played a "Lightning Bolt", I then removed my card from the field. The problem is that he didn't think that he had to discard (Liliana's Specter makes each opponent discard a card), I on the other hand thought the opposite. My reasoning is that my creature entered the battlefield before he could kill it, allowing it's ability to activate. Is he correct or am I? (I couldn't decide who was right, considering that we're both new to competitive-ish magic). Also, is the "in response to that" sentence only used for countering spell? Answers would be appreciated, thanks.