[gnunez]

Thanks for your helpful comment! I was also struggling with the bad circuits voltage divider.

[ramenbytes]

Sure thing! One of the things that helps me sometimes is re-arranging things visually.

A real divider:

  Input o----vvvvv----o-------------o Output
                      |
                      |
                      Z
                      Z
                      Z
                      |
                      |
                      o
                    Ground
  

The bad one:

  Input o-------------o----vvvvv----o Output
                      |
                      |
                      Z
                      Z
                      Z
                      |
                      |
                      o
                    Ground
  

Looking at it this way, it might be a bit easier to see that in the good circuit, both resistors can be in a Kirchoff voltage loop that includes the input, while in the bad one they can't.

(Edit: it appears I don't understand linebreaks on HN, so things are gonna appear garbled until I fix it.)

(Edit 2: Fixed.)