How can the congregation know whether a hymn is also a prayer after Mass?
First of all, let us look into the history of recessional hymns. Neither in the Extraordinary Form of the Mass nor in the Ordinary Form of the Mass is the recessional hymn a requirement. In fact, there is nothing in the General Instruction of the Roman Missal in the Mass of Pope St. Paul VI that requires a recessional hymn.
Having attended the Traditional Mass for decades, I have never seen a Mass with a recessional hymn sung. Outside of Lent, the organist plays a piece of music that elevates our thought to things on high. I have seen this in Novus Ordo Masses also.
Even though you may have heard it said that if a “recessional hymn has the form of a prayer that the priest is supposed to remain at the altar until the hymn/prayer is done.” Since there is absolutely no mention of a recessional hymn in either Mass within the Roman Rite, this must deemed as an opinion of a particular person or priest.
That’s right. There’s nothing in the General Instruction of the Roman Missal that requires a recessional hymn. Here’s what the GIRM instructs regarding the Concluding Rites:
To the Concluding Rites belong the following:
brief announcements, should they be necessary;
the Priest’s Greeting and Blessing, which on certain days and occasions is expanded and expressed by the Prayer over the People or another more solemn formula;
the Dismissal of the people by the Deacon or the Priest, so that each may go back to doing good works, praising and blessing God;
the kissing of the altar by the Priest and the Deacon, followed by a profound bow to the altar by the Priest the Deacon, and the other ministers. (§90)
The Mass for Millennials: Recessional Hymn
Now let us address your question of interest: How can the congregation know whether a hymn is also a prayer after Mass?
St. Augustine once wrote that he who sings prays twice. But we have to remember that in St. Augustine’s day, the faithful chanted the words of Scripture in Latin, either the Palms, the words of the Gospels (especially the words of Christ) or some other portion of Scripture.
Thus singing various parts of Scripture in a prayerful manner is definitely a prayer in aiding our meditation of things divine.
Further more, if a hymn is composed in such a way that one of the three basic conditions of prayer, then it would definitely be a prayer. The Litany of Saints is an excellent example of a prayer in the form of intercession.
One phrase in your example hymn is obviously a prayer of supplication: ”Send Thy archangel ...”.
St. Thomas Aquinas explains the three conditions of prayer as follows:
Three conditions are requisite for prayer. First, that the person who prays should approach God Whom he prays: this is signified in the word "prayer," because prayer is "the raising up of one's mind to God." The second is that there should be a petition, and this is signified in the word "intercession." On this case sometimes one asks for something definite, and then some say it is "intercession" properly so called, or we may ask for some thing indefinitely, for instance to be helped by God, or we may simply indicate a fact, as in John 11:3, "Behold, he whom Thou lovest is sick," and then they call it "insinuation." The third condition is the reason for impetrating what we ask for: and this either on the part of God, or on the part of the person who asks. The reason of impetration on the part of God is His sanctity, on account of which we ask to be heard, according to Daniel 9:17-18, "For Thy own sake, incline, O God, Thy ear"; and to this pertains "supplication" [obsecratio] which means a pleading through sacred things, as when we say, "Through Thy nativity, deliver us, O Lord." The reason for impetration on the part of the person who asks is "thanksgiving"; since "through giving thanks for benefits received we merit to receive yet greater benefits," as we say in the collect [Ember Friday in September and Postcommunion of the common of a Confessor Bishop]. Hence a gloss on 1 Timothy 2:1 says that "in the Mass, the consecration is preceded by supplication," in which certain sacred things are called to mind; that "prayers are in the consecration itself," in which especially the mind should be raised up to God; and that "intercessions are in the petitions that follow, and thanksgivings at the end."
We may notice these four things in several of the Church's collects. Thus in the collect of Trinity Sunday the words, "Almighty eternal God" belong to the offering up of prayer to God; the words, "Who hast given to Thy servants," etc. belong to thanksgiving; the words, "grant, we beseech Thee," belong to intercession; and the words at the end, "Through Our Lord," etc. belong to supplication.
In the "Conferences of the Fathers" (ix, cap. 11, seqq.) we read: "Supplication is bewailing one's sins; prayer is vowing something to God; intercession is praying for others; thanksgiving is offered by the mind to God in ineffable ecstasy." The first explanation, however, is the better. - Prayer Article 17