The text of the RCIA addresses the celebration of the sacrament of Confirmation on three occasions: at the Easter Vigil, at some other Mass, and at some other time outside of Mass. In each instance the rite calls for the celebrant, while tracing the sign of the cross with chrism on the forehead of the person to be confirmed, to say
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"(NAME), be sealed with Gift of the Holy Spirit" **
In my experience just the “confirmation name” is said here as in “John” or “Mary Catherine.” This can be the person’s legal first name, common first name if different, baptismal name if different, or the name of a favorite saint such as “Martin” but usually not “Martin of Tours.” Do realize however that there may be some very specific “local guidelines” to what name is to be used.
How the “confirmation name” is conveyed to the celebrant obviously will vary from parish to parish. I have seen the celebrant ask each person as they approach, work from a list, and read from a card presented by each individual or their sponsor. I also recall one instance when the each card was handed to a priest who was assisting the bishop and the priest then whispered the name in the bishop’s ear. I think that was done because the bishop was quite elderly.
**This same wording is used in the Rite of Confirmation on occasions not associated with the RCIA.