It sounds to me like you did a full rewrite by replacing the app piece by piece, sprint by sprint, releasing changes quite often and bringing that value all the way to the user. I think that is really clever.

My impression from others in this thread is that they mean "start from scratch and build until features are on-par with current product" when they say full rewrite.

Your version of full rewrite seems like it is generally applicable, but I have very little faith in the latter approach.