Templates For Pages – How to Get More Templates on the Menu

I am using WP for website development and not for blogging at all.
I notice that on my Page Design there is a drop down menu for TEMPLATE and in that drop down there is Default Template and One Column. I use One Column a lot.

My default template will show Widgets (as chosen in the widgets screen) for all Pages that use the Default Template.

Can I have more than one Default-Template-Like template that uses different Widgets from the ones I chose in the Widgets Screen per Page?

What I am aiming for is Widgets that are specific to a Page and not across the board the same for all Pages that use Default Template.

Oh, I use Themes designed by me using Artisteer and they work just fine.

WP Documentation seems to want to cross define Themes and Templates as though they were the same thing. My theme is working fine. I want to choose what widgets show up on what PAGE rather than one set for all Pages.

Thanks.. David