I think that perhaps a hint to what a vampire is occurs just a few lines before the excerpt that you included above:
She hesitated. Oskar stayed in the same position, with his hand on the jacket he had just hung up. He looked at the jacket as he asked:
"Are you a vampire?"
She wrapped her arms around her body, slowly shook her head.
"I ... live on blood. But I am not ... that."
"What's the difference?"
She looked him in the eyes and said somewhat more forcefully:
"There's a very big difference."
In the English translation, the next line is the beginning of the excerpt in the OP. I had always interpreted Eli's comments here to mean that he was not like The Man in the Wig, and he was not like the Infected woman that he had met. In mind, it ties ibn closely with Eli's comments in the Be Me A Little scene:
"I ... don't kill people."
"No, but you would like to. If you could. And you would really do it if you had to."
"Because I hate someone. That's a very big ..."
"Difference. Is it?"
"If you got away with it. If it just happened. If you could wish someone dead and they died. Wouldn't you do it then?"
"Sure you would. And that would be simply for your own enjoyment. Your revenge. I do it because I have to. There is no other way." (emphasis added)
Eli doesn't revel in his Infection. These two passages together lead me to think that in Eli's mind he is not a vampire, he is a victim of vampirism. He draws a distinction (in a twelve year old's way) between himself and his victimizer, and by extension between himself and the infected woman. Although he must do the same things to survive, he does not consider himself to be what an adult would call a sociopath, as the other two obviously are. Here then, I believe, is where we can find Eli's definition of "vampire": Someone who enjoys it.