What is the word that means there is a difference in import of the same words when the exact same words are spoken by a different person

In the case of the priest saying the words, the priest is empowered to give them special weight and significance.

From Merriam-Webster's definition of empower:

1 : to give official authority or legal power to
// empowered her attorney to act on her behalf

So:

The priest empowered the words "I now pronounce you husband and wife."
The priest was empowered to make the everyday phrase spiritually and legally binding.

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Alternatively, if it really is a word that has to start with i, then the priest imbues the words with special significance.

From Merriam-Webster's definition of imbue:

1 : ENDOW sense 3
// Spanish missions imbue the city with Old World charm
— Scott Pendleton

Endow sense 3
: to provide with something freely or naturally
// endowed with a good sense of humor

In reference to a comment posted by Shoe and a comment posted by Edwin Ashworth, the phrase you are looking for is Illocutionary acts.

This was defined by John L. Austin as follows:

I explained the performance of an act in this new and second sense as the performance of an 'illocutionary' act, i.e. the performance of an act in saying something as opposed to the performance of an act of saying something — J. L. Austin, ed. J. O. Ursom, How to Do Things with Words, 2011, Clarendon Press, Oxford.