^^ But is a DEILD done inside a FA still a DEILD?

Technically speaking, if you do a DEILD during a FA, what you are really doing is a DILD. This is because you are not actually waking up or exiting a dream, but only dreaming of doing so. However, in the process of this dreaming, and thanks undoubtedly to your excellent expectations of how a DEILD should work, you carved a path to lucidity using a DILD transition.

Is there anything wrong with that? I don't think so! It sounds like it went quite well anyway, which in the end is all that matters.

Here's a side hint, though: if you are lucid, then enjoy the lucidity! Don't concern yourself with waking up to make records of your successes; instead, concern yourself with generating more successful lucid adventures right now. There will be time later to write things down, and, if you do make the best of your lucid time, you will very likely have little trouble remembering those adventures long after you wake up for real. Also, as you may have discovered, exiting the here&now of the dream by worrying about future events like writing down the dream can be a substantial distraction that can lead to a loss of lucidity.