Yes, I remember it with all four. But I also can relate to what you're saying. My first was a 36 hour labor (it started just as I went to bed, so by 36 hours of labor, it was more like 55 hours of being awake!). I had to push for three hours. I had never been so exhausted in my life. I don't know if I remember it solely because we had it recorded or if I would have remembered it otherwise. But either way, I remember it.
Baby #2, he was a much shorter labor - 15 hours - and had shoulder dystocia and was born not breathing for a few minutes. So, hearing his cry was a relief and something I can't forget. I can't say I remember exactly how their cries sounded...just that I remember them.
Baby #3, she was 29 hour labor, exhausted, but I remember her laying on my chest crying and crying.
Baby #4, my first c-section (complete previa). She was born and they took her over to the little station thingy and were doing stuff to her. I couldn't see because the little drape thingy they use to prevent me from seeing my belly was in the way of the baby station. I finally yelled at them and they brought her over to me. She was crying and as soon as she heard me say "Hey sweet baby girl!" she immediately stopped crying and stared at me. It was awesome.
Don't feel bad that you can't remember. There is a lot I can't remember beyond those moments. I was barely awake when my first baby was born. I could hardly move from the fatigue. I couldn't hold her after for several hours due to being so weak (I also hemorrhaged and had almost passed out). My sister was there and helped. Just treasure what you do remember.