I'll prefix this post by saying that I have only been working with MySource Matrix for a week now, so go easy on me 
As a website administrator, I have received many complaints that the website's search tool doesn't return desirable results (and it doesn't). So I've searched through all the manuals I can download, and only find information on the 'Quick Search'… I have found no mention of the indexing system (if there is one) that is used by 'Joe Public' when using the search tool.
Is the index created for the Quick Search also used for the public search tool? Can someone please point me in the right direction?
Yes, they are the same index. You need to be upgraded to Squiz Client to get access to the SSV manuals, which includes the Search manual. This manual explains how to configure the Search Manager to better weight your asset attributes and metadata for more appropriate search results. If you can send me your details (via PM or email), I can verify your client status and upgrade your forum account.