Congrats on lucids!

If you were truly aware and you feel that you were lucid, it's lucid dream. As for 'dream of a lucid dream' i consider calling it like that confusing and much prefer calling it 'non-lucid about lucid dreaming actions' and those feel completely like non-lucid dreams, but your 'character' in them becomes aware and starts doing dream control and stuff, but
you don't become aware.

As for reality checks, when you do them in waking life look around and actually question reality, not just do them automatically, remember it's awareness and questioning mindset that gets us lucid, not the action itself.

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