When people in the US talk about the "short form," they are usually referring to the form that Cheng Man-Ch'ing developed. He was one of the first to popularize Tai Ch'i in the U.S., and his form remains quite popular. I believe that some people that have followed his legacy and understanding have also gone back to the longer traditional form, but preserving his flavor. The first form I learned was such a form.
Many other teachers have also created shortened versions of the traditional form and call them by various names. A few others have even created "new" long versions of the form, while still calling what they do "Yang Style." I say all this because although most experienced Tai Chi players in the U.S. will understand a reference to the "Short Form," this version of the form has been adopted only by those following Cheng Man-Ch'ing's legacy. Other Yang Style players do not really talk about a long form and a short form. Most such players would also probably care less about the sequence and number of postures than about the flavor which is given to the performance of the postures. A person doing the 103 with Cheng Man-Ch'ing's flavor will look quite different from someone performing the same postures, but with the Association's flavor.
In the Association we are urged to use the 103-movement
(also counted by some as 108 or 85) form for our general practice. As Rubén has described, we also learn and teach a 49-movement form
that is used primarily for demonstration or competition purposes. Some also practice it right after practicing the 103 in order to lengthen their practice session. We also have a 16-movement form
that I believe was designed to be used in college curricula in China and a 13-movement form
that most use as an introduction to Tai Chi for those not yet inclined or able to learn the traditional 103.
I hope this helps clarify things.