Q:

What type of software is Adobe Photoshop Lightroom? 

I have heard or thought of many terms for what Lightroom is such as image management, digital asset management, photo editing software, and the list goes on and on.  Does this package or type of software have a definitive name so I use the right term from now on? 
It would be great if someone had a direct quote from Adobe to give us a bit of proof on the matter. 

A:

The official stance is clearly to be coy with what exactly the product category is, which is why you're unsatisfied with the answers you're getting.  As Craig Walker points out, the description tag in the HTML used for Adobe's web pages about Lightroom says:

Lightroom is the leading software for digital photography editing. 

You can find this "official" text by going to http://www.adobe.com/products/photoshoplightroom/about/ and doing "view source".  Or, you can go with the more colorful description currently in the page text itself, under the heading "What is Lightroom?": 

Adobe® Photoshop® Lightroom® 3 software unites your digital photography essentials in one fast and intuitive package.  Create something beautiful.  Express your vision.  Move your audience. 
Lightroom gets you there with the tools you need to create great images, manage all your photographs, and showcase them with style and impact. 

Meanwhile, in the practical world, tech publisher O'Reilly calls it an "integrated digital photography workflow application".  Or, in Inside Lighroom from Focal Press, author and "Lightroom export" Richard Earney says (available online in the sample chapter):

What is Lightroom? 
Lightroom is an end-to-end photography workflow tool,
primarily aimed at digital photographers, but can also be used
by analog photographers who have digitized their collections.  It
is for professionals and serious amateur photographers. 

... and that sounds about right. 
Or, author Martin Evening, in a sample chapter of The Adobe Photoshop Lightroom 3 Book from Peachpit Press:

What is Adobe Photoshop Lightroom? 
Lightroom is a high-quality image processor and image database management system rolled into one, with a modern interface and fast image processing capabilities. [...]  Lightroom is not a single, monolithic application; instead, it should be viewed as a suite of application modules that combine to provide an ideal workflow for digital photographers. 

This is linked from the important concepts section of Adobe's own "Learn Photoshop Lightroom 3" page, so it's about as close to official as you're going to get — although you may also be interested in reading the elided part of the quote above for more backstory, along with related articles like The Shadowland/Lightroom Development Story. 
Scott Kelby also has a book on Lightroom from the same publisher, but in quick perusal he doesn't really bother with trying to define what the product is exactly. 
It also might be worthwhile to look at similar software.  The open-source program Darktable calls itself a "photography workflow application and RAW developer", and "A virtual lighttable and darkroom for photographers." ( Emphasis added based on Stan Rogers' suggestion in a comment below.) 
That's more useful than what Apple gives us for their Aperture product, "a professional photography application that lets you refine images, showcase your photography, and manage massive photo libraries." 
One is sort of reminded of the "It slices, it dices, it juliennes!"  advertisements in reading that, but given the success and reading the backstory, I think that's actually not unreasonable.  The intention is to cover the range of needs for photographic workflow. 

A:

Adobe itself says in its FAQ

Adobe Photoshop Lightroom 3 software is an intuitive digital darkroom and efficient assistant designed for advanced amateur and professional photographers... 

So digital darkroom is surely a good candidate, another would be photo post-production which is the "type" wikipedia puts it in. 

A:

In the Google search results, Adobe themselves call it "the leading photo management and editing software solution". ( Adobe's landing page itself is filled with marketing-fluff that doesn't actually define it.) 
As of today at least, Wikipedia calls it "a photography software program." 
I think those are about as official as it gets. 
But "official" isn't the be-all-end-all; what people actually call it is another matter, as you've illustrated.  
(Personally, I call it "photo-management software".  This seems most accurate to me.  It's primarily about photos, even though it can handle video and non-photo images.  It's not as much about editing as Photoshop, even though it does a lot of that.  It has organizational features absent in any other product from adobe.)