I've been thinking... what exactly defines a Lucid Dream?

In my last closest attempt, I was fully aware I was still in bed, I could actually still feel the bed to an extent. But I was in a dream, it was very clear, a man spoke clearly with me, and music started playing incredibly clearly, even though that wasn't happening in real life... but I could still feel my bed.

So... is a lucid dream where you concretely know you're dreaming? Or is it when you actually gain control to an extent and start applying your will? In a full lucid dream, should you still feel your bed?

I would love to call my last experience a lucid dream, but I don't want to if it isn't.

[My dream/experience]
*It’s not so much a dream, more like I knew I was lying in bed with my eyes closed seeing this, but it was a very vivid clear image and conversation, yet something just didn’t feel “real” about it. Either way, I knew I was dreaming, but it didn’t feel like a complete, whole, dream.*

I was probably flying around, but I don’t remember. I saw two dogs (small ones, a white fluffy one, and a small brown one) at a grassy park aside a big lake. As I talked to this other guy there, he went on about how the Hiroshima bomb blew up, and in the black hole it created a “scar” came out when it collapsed. I thought it was very poetic and meaningful when I heard it. While he was talking, I started hearing the song “Kryptonite” by “3 Doors Down” really vividly! I started enjoying it all and getting lost into the sense of the dream, but as I started getting deeper into the music, I felt like I was spinning, since I knew that meant I was close to a full lucid dream, I got excited and lost sight and sound of the dream!