DOC/CLN: Revise StringMethods docs
Derived from #9773, #9667. Fix docstrings to meet what current `.str` accessors does.

@jorisvandenbossche Thank you for your comments. 

Thanks to #9667, `Index` now support `.str` accessor. I've tried to describe the `Series`/`Index` return values but may not be clear because it shares the same docstring... 

CC: @mortada Can you check if you have a chance?
 @sinhrks spotted a few minor things (please see comments inline) but otherwise looks good to me, thanks! 
 @sinhrks One more general comment: you now used ```Series``/``Index``` almost everywhere, but I think in general we don't quote Series or DataFrame in our docstrings. 
Also in the docstrings here you used both forms (eg unquoted in the parameter explanations or returns section).

I don't think we have a strict rule about this (and in the tutorial docs it is a bit mixed as well), but maybe we should keep it unquoted in the docstring? Opinion?
 @jorisvandenbossche , @mortada Thanks for the comments. I've fixed the docs based on the points.

About quotes:

I also didn't have a fixed policy, only an impression that no need to quote return values because these are rendered as italic. I agree removing quote is more consistent. How should `array` be (remove or quote)? 
 @sinhrks is this good to go?
 Yes, pls check.
 Thanks! Always good to have more work on the docs
 Thanks for review! Lmk if needs any follow-ups.
