I find it interesting that lucid dreaming has an impact on your feeling of restfulness. I can't say I've experienced the same, but I'll try to reflect on this more given that almost 100% of my dream states are lucid and have been for almost all of my adult life. As an adolescent, I would lucid dream only occasionally.

My personal experience with lucid dreaming is that I am almost always aware that I'm dreaming, and most of the time I'm content to let the dream play out. Sometimes I do intervene and become an active participant in the dream, rather than letting it happen like a cut-scene in a game. And, on occasion, I will rewind a scene that I'm unhappy with or I want to revisit.

I'll also sometimes find myself in a dream location and suddenly remember all the prior occurrences that I've visited the setting. It's the weirdest feeling because I'll have what feels like years worth of memories from experiences in that dream setting that come rushing back to me. Memories that I can only vaguely grasp when I'm conscious, but as soon as I'm back in a particular setting all those prior experiences are remembered -- things that happened in the environment, character interactions, etc.

It's a bit unsettling when I wake up because for a moment I feel like I have these parallel life experiences from different realities stored in my head, mostly partitioned from each other.