<< Grrr - after 30 minutes typing an excellent reply, and clicking [Submit] the forum software said I was now logged out and lost all my hard work >>So, here goes again
As garfy said - don't worry, it can be confusing. And we all started out with similar questions.
First off, the info on this web page (from your first post)
http://www.ikmultimedia.com/news/index.php?item_id=1301 is out of date. The link near the start of the text takes me to a AmpliTube (4) Custom Shop where I can download AT4. but there is a link at then end of the text to download AT3 Fee, which I guess is the one you used.
The problem with installing AT3 Free is that it needs a Serial Number to authorise it, and, as you are new to IK Multimedia you will not have one (AT3 is from 2012).
In your situation, I would uninstall AT3 Free, run the "AmpliTube Custom Shop" installer that you downloaded. That installer installs the AmpliTube 4 standalone program and plug-ins and the free gear models. Optionally, it also installs the Authorization Manager and the Custom Shop standalone programs. And, yes, you are right, the AmpliTube software includes the gear models that affect your sound and Custom Shop allows you to try out or buy additional gear.
Run the Authorization Manager program - it may report that it is out of date. If so, let it update itself. When ready, use the "AmpliTube Custom Shop" Serial Number that you have to Authorize the AT4 software. Then you should be ready to start AmpliTube. Your free gear should be available there. If not, then start Custom Shop, log in if needed, and click "Gear" then "Restore My Gear" to make them available.
Phew!