Follow-up #15954: public API for Styler
As a reminder that we need to address this for 0.20, follow-up for https://github.com/pandas-dev/pandas/pull/15954

- [ ] Decide on public API entry (the whatsnew in the PR mentions `pandas.io.api.Styler`, but it is in practice also added to top-level namespace (`pandas.Styler`), while the API docs still use it as `pandas.formats.style.Styler`)
- [ ] Depending on decision above (which may also depend on #13634), update docs
- [ ] Would it be needed to keep back compat shim at `pandas.formats.style.Styler` to deprecate? (not sure if many people would already used this object directly instead of through `df.style`)

