One clue that it might not be is that it refers to "Microsoft Office Starter 2010," whereas Microsoft calls it "Microsoft Office 2010 Starter."
Not correct. It's called "Microsoft Office Starter 2010"
From the product itself..

I see the version I have is 14.0.7128.5001 and **The most recent Click-to-Run release is version 14.0.7130.5000**released
on Tuesday, August 12, 2014.
The updates are supposed to be installed automatically but this one wasn't and instead I and others have been seeing this pop-up "click to run" box instead.
I agree it's uncharacteristic of MS to do an click-to update this way but it's certainly not unheard of.
If anyone who's already installed this can verify which version they now have it would be helpful.
All I'm looking for is verification from someone at Microsoft whether this in fact is from them and that it's alright to go ahead and install it.