wp_add_inline_style() is intended to add additional CSS to an existing stylesheet. The idea is that you may need to dynamically alter the stylesheet – for instance, you have user-selected colours associated with categories and you would like to colour each category’s title accordingly. You can’t hardcode this since the colours in question are not known, but live in the database.
So
wp_enqueue_style(
'custom-style',
get_template_directory_uri() . '/css/custom_script.css'
);
is just queueing an existing stylesheet to which we’ll attach our dynamically generated CSS.
It seems what you’re trying to do is is load user-entered CSS. I wouldn’t really recommend this – offering tweaks via the customiser is a good way of allowing users to personalise the theme, but if they need to make substantial changes, what they’re really after is a child theme.
However, if you are to do this, I would use wp_enqueue_style() to load the theme’s style.css and then attach customisations to that. (Most themes seme to hard-code the style.css into the <head>, but I don’t know hwy).
Developers: You should be aware of this trac ticket: http://core.trac.wordpress.org/ticket/24813