In both the Yagna Shaala archanai and a temple archanai, once the priest receives a request, he proceeds to do what is called a "Sankalpam". The priest asks the bhaktha to introduce himself: "who is initiating this session of praise for the Lord and what is the purpose". The Bhaktha dutifully answered their name and purpose. [Interesting side-note is that in the last 20-30 years Brahmins have names that are very different from their 'Dasya' name - a name kept during Upanayanam. I have seen some very elderly priests get irritated and ask "Rajesh'a evanda avan 'rajesh'? Nithyasoori peyara idhu?". and it is hilarious to see the confusion and terror in Rajesh's face as he tries to figure out what is being asked of him. I have been in that situation and had to be helped out]. Since today's bhakthas don't have a clue as to why they are initiating an Archanai, the priest does a "Recommended Settings" Archanai with "Dheerya, Dhairya, Vijaya, Arogya, Aishwarya .... abhiVridhYartham" as the purpose.
Observations on the answers to the "who is initiating the archanai" is interesting. I recently saw an elderly 'vidhvith' (who was substituting as a priest) get totally confused by an answer and had to be told what it meant. And the consequent thought-train is what led me to this post. It is hard to imagine what a priest's assessment would be of the differences between the 'theory' they learnt in school vs what is actually in practice. But we can always hazard a guess. In a modern-day temple, the priest puts a plate in front of the Bhaktha and says "peyar naksathram sollungo" (introduce yourself). In the last few decades responses such as "Perumaal peyarukke pannungo" or "Saami peyarukke pannidunga iyere" etc have started coming from the Bhakthas. After the incident referred above, I realized after mulling about it for a few months that such a response really may not the answer the question. The archanai or praise was always going to be for the God. There was never any question about that. Unless the Bhaktha presumptuously thought there was a possibility that the priest would sing 10/28/108/1008 words in the Bhaktha's praise, he need not remind the priest who this archanai is for. The question was always "who is initiating this archanai". Answering it as "God" is kinda vague/stating the obvious/being recursive unless the bhaktha is deeply philosophical and is indicating that his 'antharyamin' is the trigger. It is a totally different question whether this Bhaktha does not want to gain any benefits out of this. Stating that to the priest would save the "dheerga... dhairya.." etc that follows.
I guess my train of thought was along the lines of the amount of variations we have internalized over time, that we don't even know about. It would be an interesting exercise to see what would happen if a priest was transported to this day and age from (lets say 1250 A.D.) or - if we were transported back to that time.