Ok, I'll bite.
You are correct that it does mean, among other things: "
to stammer; hence,
to babble; to use vain repetitions"
Your article also says:
"But does Jesus mean to exclude the possibility of devotions like the Rosary or the Divine Mercy Chaplet which repeat prayers? No, he does not. This becomes evident when in the very next verses of
Matthew 6, Jesus says:
Do not be like them, for your Father knows what you need before you ask him. Pray then like this: Our Father who art in heaven, Hallowed be thy name. Thy kingdom come. Thy will be done, on earth as it is in heaven. Give us this day our daily bread; And forgive us our debts, As we also have forgiven our debtors; And lead us not into temptation, but deliver us from evil. For if you forgive men their trespasses, your heavenly Father also will forgive you; but if you do not forgive men their trespasses, neither will your Father forgive your trespasses.
Jesus gave us a prayer to recite! But notice the emphasis on living the words of the prayer! This is a prayer to be recited, but they are neither “empty phrases” nor “vain repetitions.”
I disagree in that it is a "model". Because in Luke 11:1-18 we have the same thing repeated.
Lord, teach us to pray, just as John taught his disciples.” 2 He said to them, “When you pray, say: “‘Father, hallowed be your name, your kingdom come. 3 Give us each day our daily bread. 4 Forgive us our sins, for we also forgive everyone who sins against us. And lead us not into temptation.’
And even your own article warns:
"This does not mean that there is not a danger of mindlessly repeating the Rosary or other such devotions. There is. We must always stay on guard against that very real possibility."
I am aware that I left the last sentence off purposely because as a Baptist, I do not accept:
"the great devotions Holy Mother Church provides for our spiritual nourishment."
John Gill comments here:
"After this manner therefore pray ye,....
That is, in such a concise and short way, without much speaking and vain repetitions; making use of such like words and expressions as the following:
not that Christ meant to pin down his disciples to these express words, and no other; for this prayer is not a strict form, but a pattern of prayer, and a directory to it, both as to brevity, order, and matter; for we do not find the disciples ever making use of it in form; and when it is recited by another Evangelist, it is not in the selfsame words as here; which it would have been, had it been designed as an exact form. Besides, Christ does not bid them pray in these very words, but "after this manner"; somewhat like this: not but that it is very lawful to use the very express words of this prayer in any of the petitions here directed to; and which indeed were no other than what good people among the Jews did frequently make use of; and which were collected and singled out by Christ, as what he approved of, in distinction from, and opposition to, other impertinent expressions, and vain repetitions, which some used; as will appear by a particular consideration of them.
Our Father which art in heaven.
This may be looked upon as the preface and introduction to the prayer, and regards the object of it, and his character, which is an epithet of God, often to be met with in Jewish writings, and particularly in their prayers; for thus they (k) say, "Mymvbv wnyba, "our Father which art in heaven", show mercy "to us, because thy great name is called upon us."
Again (l), let the prayers and the requests of all Israel be received by , "their Father, which is in heaven". They seem to have a regard to this prayer, when they apply that passage in
Proverbs 3:35 "shame shall be the promotion of fools", to the nations of the earth, who, they say (m), "do not consider the glory of the law; and how, say they, "our Father which art in heaven", hear our voice, have mercy on us, and receive our prayer?''
So in confessions, thanksgivings, and sacrifices of praise, they required, and looked upon it, as the main thing, for a man to direct his heart , "to his Father which is in heaven (n)."
By "father", our Lord means the first person in the Trinity, who is the Father of all men by creation, and of the saints by adoption; who are to address him in prayer under the character of "our Father", partly to command a reverential fear of him, and partly to secure boldness and liberty of speech before him; and also to express fiducial confidence in him, faith of interest in him, and relation to him; which arises from some experience of his paternal love, and requires the witnessings of the Spirit of adoption; and inasmuch as the direction is not to say "my Father", but "our Father"; it shows that we should pray for others as well as for ourselves, even for all the dear children of God. It is a rule (o) with the Jews,
"that a man ought always to join himself in prayer with the church;'' upon which the gloss says, "let him not pray the short prayer , "in the singular, but in the plural number", that so his prayer may be heard.''
The object of prayer is further described by the place of his residence, "in heaven"; not that he is included in any place, but that the heaven of heavens is the place where he most eminently displays his glory: and this may teach us to look upwards in prayer, and seek those things which are above; and also, that this earth, on which we dwell, is not our native country, but heaven is, where our Father dwells.
Next follows the first petition, hallowed, or sanctified be thy name; so the Jews (p) in their prayers, "Kmv vdqty, "let thy name be hallowed", or "sanctified by us", O Lord our God, before the eyes of all living.'' And very often (q), "let his great name be magnified and sanctified in the world, which he hath created according to his will.''
Source
John Gill also comments on Luke 11 saying:
"The doxology there used, and the word "Amen", are here omitted. Some of the petitions in this prayer are not delivered in the very same words as they are in Matthew. The three first petitions are word for word the same; for though the third petition is different in our translation, it is the same in the original. The fourth and fifth vary: in Matthew the fourth is, "give us this day our daily bread"; here in Luke, "give us day by day our daily bread." The fifth in Matthew is, "and forgive us our debts, as we forgive our debtors"; here, "and forgive us our sins, for we also forgive every one that is indebted to us".
And these verbal variations, though the sense is the same, together with the omission of the doxology, and the word "Amen", show, that this prayer was not designed to be an exact form, and to be so used, but as a directory of prayer."
Source
And personally, in the 40 years I've been a Baptist, I can only remember a handful of times, perhaps 4 or 5, where "The Lords Prayer" was recited in church.
But here we are, back to where we started.
God Bless
Till all are one.