I discussed this with a canon lawyer, and we agreed that your interpretation is correct. You do not need your bishop's permission to bring a particular legislative text to the attention of your prayer group; on the other hand, members of the prayer group certainly have a right to go to the diocese and ask if your interpretation of the document is correct.
Canon 1003 §1 of the Code of Canon Law states: "Every priest, and only a priest, validly administers the anointing of the sick." The fact that only priests can give the anointing of the sick was infallibly defined at the Council of Trent (Canons Concerning Extreme Unction 4).
While it is impossible for anyone but a priest to administer the sacrament of holy anointing validly, this does not in principle rule out the possibility of non-priests, including laity, administering non-sacramental anointings. In principle, it is possible for non-priests to anoint people with oil and ask God to bless or heal the person, while acknowledging that this is not a sacramental anointing and that God has not promised to use this anointing the way he has promised to use the sacrament of anointing. However, such anointings would run a risk of confusing people: (1) They might think that this was a sacramental anointing; (2) they might start relying on this easier-to-obtain anointing rather than on the sacrament; and (3) they might start using this as a way of "getting around" the cautions and regulations the Church naturally has had to set up to protect the sacrament of holy anointing, just as it has to protect all the sacraments.
To avoid these problems, the Church might prohibit non-sacramental anointings, just as it might prohibit a near-imitation of any sacrament. Suppose people were taking catechumens and building them into a state of prayerful expectancy and then pouring water on them and saying, "May the Lord regenerate you and give you new life and forgiveness of sins and open unto you the gates of heaven, in the name of the Father and of the Son and of the Holy Spirit." This would be a close imitation of baptism, and grave problems would result if it were not made absolutely clear to the catechumens that this is not the sacrament of baptism, that God has not promised to use this ritual to give them any of the blessings of baptism, that they are required to go ahead and seek baptism after this, and that this cannot be used as a way of avoiding baptism or the Church's regulations concerning it. The Church might well choose to prohibit all such non-sacramental water rituals.
In our day, especially in charismatic circles, there are many people who encourage prayerful expectancy in those who seek healing and then perform a non-sacramental anointing on them. Because of the problems this widespread practice may cause, the Church now has prohibited non-sacramental anointings. The November 1997 Instruction on Collaboration states:
The non-ordained faithful particularly assist the sick by being with them in difficult moments, encouraging them to receive the sacraments of penance and the anointing of the sick, by helping them to have the disposition to make a good individual confession as well as to prepare them to receive the anointing of the sick. In using sacramentals, the non-ordained faithful should ensure that these are in no way regarded as sacraments whose administration is proper and exclusive to the bishop and to the priest. Since they are not priests, in no instance may the non-ordained perform anointings either with the oil of the sick or any other oil. (Instruction, "Practical Provisions," art. 9 §1)
This does not say simply that the laity cannot perform the sacrament of holy anointing. It says that they may "in no instance . . . perform anointings, either with the oil of the sick [used in the sacrament of holy anointing] or any other oil." The purpose of this is to protect the sacrament of holy anointing from being confused with these other anointings. Since there is the danger of confusion in our age (which is pretty confused, anyway), it is simpler to just prohibit non-sacramental anointings rather than run the risks of people neglecting the sacrament Christ did institute for this purpose and putting their trust in a different ritual.
Note that the document also stresses that when administering sacramentals, the laity must ensure that the sick realize that these are not sacraments. That applies, of course, to sacramentals approved by the Church. When it comes to prayer with anointing, which is not an approved sacramental, the laity are prohibited from doing it altogether. They may perform only those sacramentals that the Church has approved for them to perform.