You have taken a different path. May I ask, what your goal is?
In your first Circuit:
Of course, in any Transformer, two Coils are needed!
In any "Generator" a Stator Coil and moving Source magnetic Field, either another Coil or Permanent Magnet are needed.
What you show, a saw tooth waveform, is not how we do it, we do it in the aim to "Generate" excess Energy.
We have many threads on our forum explaining how this can be done, so I am at a loss to see where your approach is directed.
Your second circuit is much better, but the waveform is not quite right yet, it is a saw tooth, but not yet right.
I watched your video, Great Work! Thanks for Sharing! You are much closer on your second circuit!
Pay special attention to the Polarity's:
Top marks for sharing! Thank You! Sorry if this post comes across as grouchy, I don't mean it that way, been a long day. With a bit more work on your second circuit, you will get it, just a bit more work!
@CD would you like to extend this further?
Best wishes, stay safe and well,