Installing the programs (like trial games) writes entries into the Windows Registry. Those entries are used to by Windows to recognize that a program has been installed. Meaning you click the program's launch button and the program will run. If those entries are missing then most programs will not run.
Trial versions of software also writes some type of info which indicates the date the program (game) was installed. Once the trial period is over, then that's it; the end. It doesn't matter which site you downloaded from because it is the same trial program.
If you want to reinstall the game, then you must edit the Windows Registry to eliminate the game's entries. Easier said than done especially if you do not know what to look for. Deleting the wrong entries can have one or more of the following effects:
1. A different program or basic Windows function no longer works properly.
2. Your system becomes unstable
3. Your system will not boot.
If you don't know what you are doing then don't mess with the Registry.
You have three options:
1. Stop playing that game.
2. Buy that game.
3. Wipe your hard drive and reinstall Windows and any other programs you want and that trial game. When the trial period ends and you still want to play again, then repeat this step.