What are the differences and considerations between "the position of the third generation of the clan" and "the position of enshrining the third generation of the clan"?

There is no difference in meaning between the two. Whether to add the word "enshrine" or not depends on local customs and habits.

"Worship" here means to offer sacrifices to deceased ancestors to show that one does not forget the ancestors and to express one's filial piety, which is also a traditional etiquette. The two writing methods "the place of the third generation of the clan" and "the place of the three generations of the clan" are written on the tablets of the deceased clan members, and placing the tablets respectfully on the altar certainly has the meaning of "enshrining". . Therefore, the two ways of writing have exactly the same meaning. Whether to add the word "enshrine" or not depends on local customs. How the previous generation wrote it should be written in the same way as this generation. However, in most parts of the country, the word "enshrine" is found at the beginning of ancestral tablets.