Simply, a personal signature on the one single form of Ti or Toudi that is Okinawa Ti. Ti is the term used to designate and denote a singular form of empty handed defense that today is referred to as karate, i.e. formally China-hand and currently empty-hand. The various styles and systems such as Isshinryu, Gojuryu, and Shorinryu, etc. are those designations given by individuals who have reached a level of mastery in Ti/Toudi that they feel the need to name their own personally formed system that is ti/toudi but with a personal signature or essence that makes the distinction that this person is responsible for this way of practicing, training and using Ti/Toudi.
In a nutshell, my view, is this is merely a form of ego driven recognition and in modern times a means by which one differentiates a teacher and style from others with promotions as to a particular personality of the person and style as unique thereby bringing more students and income into the dojo. Cynical, yea a bit but with a smidgeon of truth.
Lets look at this a bit historically. At first there was only Ti/Toudi. Then to distinguish it was to separate it from one another as to location ergo why Ti took on names such as Tomari-ti, Shuri-ti and Naha-ti. To distinguish is actually a need for man to differentiate and make unique something that was previously shared by all.
As time passed and as it became more available to all Okinawans there came a further need to differentiate, distinguish and make unique the jutsu of ti into systems/styles that were associated with individual masters of long standing especially since the three main villages slowly were absorbed into many that is Okinawa, i.e. Goju-Ryu, Shorin-Ryu, Uechi-Ryu, Isshin-Ryu, and so on.
Even today in modern times those original "styles" soon branched off into more unique styles and systems associated with new and more prolific masters to differentiate between just karate and the special karate's that you would need to fight and defend or worse compete. There are not a long list of variations within a system like Shorin-ryu, i.e.
Shōrin-ryū Reihokan[8]
Shōrin-ryū Shidōkan normally called Shidōkan or Okinawan Shidōkan
Shorinkan
Kobayashi Shorin-ryu
Shōrin-ryū Seibukan
Okinawa Seidokan Shōrin-ryū normally called Seidokan
Shōrin-ryū Kyudōkan normally called Kyudōkan
Koporyu Shorin-ryu
Oshukai[9]
Okinawa Shorin-ryu karate Shinkokai
Chubu Shorin-ryu[10]
Matsubayashi-ryu (also translates to Shorin-ryu)[10]
Shorin-ryu (Shaolin)[10] also known as Shobayashi.
Ryukyu Shorin-Ryu[10]
Rendokan Shorin-ryu
Seibukan Shorin-ryu
Matsumura Seito Shorin-ryu
Even in the singular style/system of Isshinryu there are many different factions of that system yet to be renamed into new more individualize unique systems/styles except those who merely added to the name Isshinryu, i.e. Advincula's Isshinryu, Long and Wheeler's Isshinryu, and Nagle's Isshinryu - all three practiced and taught differently with small variations, etc.
In the end there is absolutely nothing wrong with this as long as the practitioner/participant knows this and understands that in reality there is only one system of karate called "Ti" and that all of them are fundamentally the same except for that unique signature on Ti used by the dojo sensei for teaching, and often renumeration sake, today's karate of Okinawa. In the end all systems and styles of martial systems rely on the same exact fundamentals, the fundamental principles of martial jutsu or in this particular case, "Ti."
Addendum dtd Friday February 21, 2014 at 15:15
It is also stated in an article written by an Okinawan karate master that his belief is that styles were adopted by Okinawans due to the historical influences of Japan's martial systems that rely heavily on a hierarchal system, etc. Because Okinawa was absorbed by Japan and Japan dictated this along with influences through the interactions of Okinawan masters and Japanese martial arts, i.e. Funakoshi Sensei, etc. this came into being, i.e. styles and systems.
Ti is Ti or Toudi. The principles of martial systems are the concrete foundation of all variations of empty handed Asian combatives. Putting a unique stamp on a personal version is just that, a symbol or stamp or title to differentiate a personal belief and view of a singular system of Okinawan Toudi.