I am looking for a search-engine API, preferably in PHP, that I can use to build a custom search engine that only searches my website.
And whose results I can style with CSS, so I believe that requires the results page to be in my site / on my server.
I would prefer to not use Bing's API.
But, if you have any APIs or solutions to achieve what I am trying to achieve, I would sincerely appreciate it and feel free to share them.
Use Elastic Search, it is dead easy to set up and use because it's just REST. There is also already a very well done php lib for it, Elastica.
You'll have to build your own index based on whatever you need. But I can tell you that it is very efficient and we implemented it within a day.