It sounds like you are getting extremely close.
If you understand that you are in a dream then it is a lucid dream; however if you try to act as if it is a lucid dream, but still don't really understand for real that it is a dream, then it is usually called a "False Lucid Dream".
It sounds like you were having a dream like that.
A "dream within a dream" is not actually multiple "layers" of dreams, they are just different dreams that happen one after another.
You can have a dream, then get another dream that happens to start in a dream bed, and this gives you the illusion that you "woke up" from a "deeper" dream, but they are really just two different, separate dreams.
Anyway, you are getting very, very close.
You shouldmake it a habit to always double-check your surroundings every time you wake up in the mornings, because this will make you start doing the same checks in "dream awakenings" as well, and that's when you have a wonderful chance to become lucid!

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