I've been doing some more digging since posting this request. I found that there is a static set in /sapphire/search/SearchForm.php
protected $classesToSearch = array("SiteTree", "File"); So it looks like it searches the Files table as well as the SiteTree.
In the same file there is also the following method:
* Set the classes to search.
* Currently you can only choose from "SiteTree" and "File", but a future version might improve this.
function classesToSearch(array $classes)
So it looks like having created a new SearchForm object you could call this method in your search method (probably in Page.php) and remove Files - by just passing in SiteTree..
In this method there are also arrays set up of which columns to search. However there doesn't seem to be a way to change this other than hacking it.
I cannot find any useful documentation on how the search works, the tutorial is a start, but more is needed.
Searching the File table is not very useful as it only holds the file names/paths. What would be useful is to be able to search the assets if they are something like pdf files.
I have come to the conclusion that SilverStripe's search is not one of it's strengths and I would really like to see it improved considerably.