It is true the docs are quite confusing at the moment.
There are 2 version of Google Custom Search: free (CSE - Custom Search Engine) and paid (GSS - Google site Search).
When using the free one you have 2 options: + implement javascript widget with search box and placeholder for results: https://developers.google.com/custom-search/docs/element - this is a standard way of implementing CSE + use JSON REST api - https://developers.google.com/custom-search/v1/overview - this is experimental feature and has a limited free quota (100 queries as of now)
When using GSS (paid one), you can still use javascript widget, but additionally you get access to XML results: https://developers.google.com/custom-search/docs/snippets#presentation_layer
Ususally first you create a search engine in the Control Panel: http://www.google.com/cse and then you use either js widget to access it or JSON (free but low quota) or XML (paid) feeds via HTTP GET requests from your website.