The beta version of Dominion: Storm Over Gift 3 was released as a demo (v1.0), and then an updated version came up as well. There are 5 single-player missions and a couple multiplayer maps.
Dominion: Storm Over Gift 3 Public Beta v1.0
Dominion: Storm Over Gift 3 Public Beta v1.1

Additionally, a trailer of the game was released prior to its publishing, which is available here.

How is it a beta if it's 1.0? Where does it say it's beta?

There's a beta number in the bottom left corner of the main menu screen, and it says "beta" both in the installer and the accompanying readme files. I suppose that the v1.0 and v1.1 thing refers to the demo release, which was based on a beta build of the game, so technically it's still a beta.

Version numbering can sometimes get quirky. For example, if you check out the Command & Conquer: Red Alert demo, the version number will say something like v1.20, whereas the first release of the full version is v1.04, and there has never been any v1.20 release of the full game (neither through patches nor a stand-alone release).

BTW, one of the Warcraft II demo releases is called "WARCRAFT 2 DEMO BETA v1.02elc". Talk about weird version numberings (elc probably stands for "electronic distribution").

You are almost right. The meaning of "elc" can be traced back to the earliest known screenshot of Blizzard's website:

It's roughly dated to January 1996.

So, "elc" obviously means "Electronic Demo", which is what other companies called "Internet Demo".

About version numbers:
The Warcraft 3 beta (just as an example) had version numbers beginning with 1.00 . The retail version started with 1.00, too, but the build numbers were higher. (the last build number of the beta was seemlingly manually changed: version 1.34 has build number 9001 instead of something around 4430. maybe a reference to dragonball ).

Similarly, the downloadable demo of Starcraft has "ED" in its version number, whereas the (enhanced) CD version is v1.01CD.