In this context: no. It's just the astronomical description (by convention) for the 180 degrees of longitude "left" of Greenwich. But sometimes I have problems with "western" designating a cultural/political unity where Australia makes part of whereas Brazil might not (depending on context).
And indeed: when, actually, is Summer 2011 in Australia?
What's wrong with "The Western Hemisphere"? The world's a sphere (roughly), so any way to denote longitudinal halves is going to be arbitrary.
Are you saying that we just shouldn't use hemispheres at all (save maybe Northern and Southern because they're not arbitrary) and instead say things like "North and South America" or "Europe, Africa, Asia, and Australia"?
I do agree that it's not a great a little awkward because the very western parts of Europe and Africa are technically in the Western hemisphere. But it is a much shorter way of denoting things than listing continents.
For those of us not living in the US, or not wanting to stay up late, or even for those that want to work out the best time to go out to look (if it's cold where you are), http://www.stellarium.org/ is a free cross-platform planetarium tool that allows you to simulate the skies from your personal location and gives you loads of info to boot. You can also switch dark skies mode on if you install it on a laptop and take it outside without it destroying your night sight.
If you want to see what a lunar eclipse would look like from the moon's perspective, Celestia (http://www.shatters.net/celestia/) is a free cross-platform solar system simulator where you can go to the moon, set the time accordingly and watch the sun go behind the earth.
To be clear for those on the west coast, it's actually 12/20/2010 starting around 10:30pm. The "morning" is for the eastern seaboard where it'll be 3am for totality, whereas for those of us in the west it's 12am.
And I'm from Australia so Summer 2011 is a difficult thing as well.