When I do a site-specific search on google.com:
site:http://one-month-of-chat-logs.github.io security
I get 12 results. I signed
I found that it is impossible to get the right results using Google APIs. Even if the search is only for one website, their search results are different if you use their UI vs use the API and pay for it. This is I guess, because google makes more money if they can show ads, while APIs are definitely only a face saving measure.
Since some of you are ok with a paid solution(@ihsan) you can try using a third party service like https://www.expertrec.com where you can control your crawl (so crawl depth is not a problem), ranking (adjust it the way you like), use the API or the full solution, with out any ads.