The members of the Sri Vaishnava sect of Hinduism (of which I'm a member), especially the Iyengars (Sri Vaishnavas who are Brahmanas), are divided into two sub-sects, Thenkalais and Vadakalais. These sub-sects both agree on the Visishtadvaita philosophy of Ramanujacharya that characterizes the Sri Vaishnava belief system, but they have several fine-grained but important doctrinal differences; this web page does a good job of laying out the points of dispute. (See my answer here for more details.) In any case, these differences don't have much impact on the day-to-day religious practices of Iyengars, but they do have some: for instance, Vadakalais wear a U-shaped namam on their foreheads while Thenkalais wear a Y-shaped namam, Vadakalais bow down twice while Thenkalais only bow once, etc.
But my question is about another such difference, concerning the words used in the Sankalpam. Before you do any Hindu ritual, you are supposed to begin by saying the Sankalpam, a formulaic utterance where you tell the gods where you're located in the universe, the time and date you're performing the ritual, what ritual you'd like to do, etc. Hindus of all sects say the Sankalpam in almost identical fashion, except for a single line. (See page 2 of this PDF). The line begins with the words "Sri Bhagavat Aagjna", meaning you are doing this ritual with the blessings/approval of Bhagavan or God (similar to the Latin expression Annuit Coeptis). But how the line is continued depends on what sect you belong to.
If you're a Shaiva or Smartha, you would continue the line by saying something like "Parvati Parameshwara Preethiyartham" or just "Parameshwara Preethiyartham", meaning that you are doing the ritual for the sake of Parameshwara (a name of Shiva). But if you're a Sri Vaishnava, what you say next depends on what sub-sect you belong to. Vadakalai Iyengars say something similar to the Shaivites: "Sriman Narayana Preethiyartham", meaning that you are doing the ritual for the sake of Sriman Narayana (a name of Vishnu). Thenkalai Iyengars, on the other hand, say "Bhagavat Kainkarya Rupam", meaning that your performance of the ritual is a form of service to Bhagavan (God).
My question is, what is the philosophical difference between Thenkalais and Vadakalais that could possibly lead to such a small change in wording? Thenkalais and Vadakalais are all Vaishnavas, so presumably they'd all agree that Vishnu can rightly be called Bhagavan, the supreme being. So what's the difference they're quarreling about? Doing something for the sake of a god vs doing something as a form of service to a god seems like splitting hairs to me. Could it be that I'm mistranslating the Sanskrit words Preethiyartha and/or Kainkarya? Or am I missing some subtle philosophical distinction here?
On a side note, I just noticed that the preamble of the Vishnu Sahasranamam contains the phrase "Sri Mahavishnu Preethyarthe". Could that be related to the Vadakalai practice of saying "Sriman Narayana Preethyartham"? If so, why would they substitute the name "Mahavishnu" with the name "Sriman Narayana"?