This is a permission problem and you have to check the owner and group of the files and compare them with the com_multisites directory. In general the problem come from the fact that the owner and groups are different.
You can also try to use unix chmod to increase the files and folder permission.

But what if ALL of the following (folders and included files) are owned by apache with 777 permisions

administrator/includes/defines.php
administrator/components/com_multisites/*
components/com_multisites/*

Then what could be causing

* Unable to write entry
* Error during installation of the file "administrator/includes/defines.php"
* Copy failed
* Copy failed
* Copy failed
* Copy failed
* Copy failed
* Copy failed
* Copy failed
* Copy failed

Perhaps this is apaches and 777 but I guarantee you this is a permission issue and that you can not modify the files.
So you have to check again the problem of permission.
Perhaps this is an open_basedir that does not give access to the file or another setting somewhere that does not allow to modify the existing files.

Not only the files but also the parent directory must have the permission like the "public_html".