EDIT: So, after re-reading your original post I've decided that my first answer was completely wrong, but I left it below mainly 'cuz I don't think I've ever said that much in one post before.
What you want to do is follow the instruction on the first post in the link you gave. Basically, you need to download TWO items: the self-installing update 1.66, and then the update to 1.93. Note the colors I just used, they are the same color as the links for the files you need to download from the link you posted.
Good luck!
Below is my first post (Please disregard this as anything meaningful):
It sounds to me like you might have all the files you need, just in the wrong places.
First off, find the rules file for the version of the mod that came with your game. This file will either be a .bic .bix or .biq file. These types of files determine what version of Civ3 the mod is for: .biq files are for Conquests, .bix files are for PTW, and .bic files are for the original (which we call vanilla) Civ3.
Now, once you know what version the mod you have is, you can figure out where to put the files. In the main Civ3 folder on your computer there are sub-folders for PTW ("CIV3PTW") and Conquests ("Conquests"). In each of these folders, there is an additional sub-folder named "Scenarios". This folder is where you want to put all the files for the mod you are trying to play (not just the rules file). These other files will go in a sub-folder with the same name as the rules file, located in the "Scenarios" folder.
Probably, you have version of Test of Time that came with PTW. This means that the files the game is missing need to be placed in the "Civilization III\CIV3PTW\Scenarios\TETurkhan\" folder. Specifically where those files go in this folder is a bit more complicated, but usually patches come with the proper file structure to make it work.
So try this: take all the files and folders that you unzipped from the patch, and put them in the "TETurkhan" directory. It will probably ask you to override some files already there, and this is ok. If it doesn't ask you that, check to be sure you only moved the files in the patch, and not the folder that you unzipped them into. For example: if you unzipped the patch into a folder called "Patch", you don't want the "Patch" folder inside the "TETurkhan" folder. You want the files inside the "Patch" folder to go in the "TETurkhan" folder (they will probably have names like "Art" and "Text"). Now, if you haven't already, take the rules file from wherever it is now and put it in the "Civilization III\CIV3PTW\Scenarios" folder.
If you still have problems post them here and hopefully I or someone else can help you. Also, it will help us to know what version of Civ3 you have (It sounds like you have Conquests or Civ3 Complete), and also what specific version of the Test of Time patch you downloaded.
