A false awakening is when you wake up in a dream but do not actually wake up, so this would count as one. Seeing you are still sleeping and dreaming but have only dreamed that you woke up it is still possible for charachters to re-appear. If you want to become lucid I can imagine that seeing those will trigger a reality check and will make you realize that you are still dreaming (seeing FIs usually make lucidity go away).

Going by what you've written it sounds as if you became lucid a couple of times already (I envy you! ) which is great. As for your dreams being weird: I don't think they are. When you are dreaming your system for logic is impaired or does not function (or so I've been told) hence a lot of strange or impossible things are in fact possible. Dreams don't haev to make sense, in my opinion. As long as they are fun to us it's all cool.