Yes, it is the original article, regardless of length, but some publications don't require the actual original.
The term "clip" is carried over from before there were photocopiers, and it was an actual copy of the article in question, clipped or cut directly from the magazine. Some publications still want to see the full original article -- especially since it's so easy to "create" a clip using word processing programs. The slick glossy paper ensures that it's an original piece (or at least cuts down on the chances it's faked). Generally when I get a published article, I request five copies of the magazine for my "tear sheet" aka "clip" file, just in case I need one on short notice. I was once required to give an original clip to a product manufacturer when I was attempting to interview their R&D department head, to prove I was a published magazine writer in the field I was claiming.
If you don't have a clip with your name on it, one satisfactory method (such as something written for a newspaper with no by-line) is to get a letter from the editor of the paper stating that you were a staff writer and that the articles in question are yours.
While it may seem extreme for publications to be asking for this, it too often happens that people will try to fake their credentials and then the publication is stuck holding the bag (and if this didn't happen, there wouldn't be a stink about teachers trying to get promotions with "diploma mill" certifications!) What bag would they be holding, you might ask? If they write a good article, did it matter whether the person faked prior credentials? Yep. Someone who took the time to fraudulently state their credentials is also likely to fake the quotes and the original content, something that could get the publication sued.
Several writers organizations also require original clips accompanying an application to gain membership (Outdoor Writers Association of America - OWAA is one of them.)
JMHO, of course!