...is not optimal in the docs because the leaving bars are all kept.

Also, the name Solver could be misleading. Substep (completion percentage) could be better.

A new verbosity level, 3, or 20, could be used to deactivate the Newton progress bar while showing the Step progress.

But to make it clearer, a dict could be used.

None, False/0 = no output
True/1, 11, "bar+bar" output
2, "text", output
10, "bar" for steps only
12, "bar+text" output
20, "text" only for steps
21, "text+bar" output

02 could be problematic (no integer) but is not really useful anyway.