"Very Similar" vs "Based On" for purpose of paper "narrative"
In an ordinary scientific paper, nobody expects to get a narrative of your investigation, but rather a clear explanation of how the ideas fit together.  In fact, it has been argued that success in science is precisely defined by the separation of statements from the circumstances of their creation.

I would thus argue that you should no more feel obligated to tell a reader that you invented an idea independently than that you invented it while riding a bicycle.  You may not have started by basing your method on the other method, but now that you know it exists, you can skip all sorts of argumentation and verification by changing the way you think about your method in order to base it on the prior method and the work that has been done to establish it.

Thus, I would recommend using something like your statement 1a/1b.  That is not the history of events, but based on what you report, it appears to me to be a true and more accurate characterization of the current relationship between the concepts.  Since you discovered this prior method, you are no longer proposing a new method: instead, you are applying a slight variation on an existing method, as adapted to your circumstances, and should report that fact accordingly.