Vanity can also be about impressing one's self.
I don't believe though, that repetition is being categorically separated from its modifier within the context of prayer.
Think about. If you asked me the same thing ten times in a row, I would have reason to doubt your ability to reason being in control of your need to obey.
Repetition is a good thing generally, but too much of it becomes harassment.
This proposes a model of prayer which I feel focuses too mich on prayer as a petition, when this very frequently is not the case at all.
We are commanded to pray without ceasing.
Now on the subject of vain repetitions, our Lord was clearly referring to the glossolalia engaged in by members of various Pagan religions and also some Gnostic cults. There is a recovered text of one of the Mithras mysteries which contains an example of this; a prescribed sequence of syllables that meant nothing in any language, usually vowells, thought to be some kind of password.
The Mithras mysteries and Gnosticism were the main rivals to Christianity during the first two centuries.
Now, praying repeatedly the Lord's Prayer, or an "arrow prayer" like Kyrie Eleison or the Jesus Prayer, cannot possibly be vain repetition because the request is not in vain, and the prayers are not offered for the sake of vanity.
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To the OP I would suggest the Philokalia. This is the definitive guide to unceasing prayer, and one can find a text of it online. However, it is written with a monastic audience in mind. Another valuable book on this subject, which you can obtain via iBooks, is "On the Prayer of Jesus" by St. Ignatius Brianchaninov, the celebrated 19th century Russian hierarch and ascetic.
Alas, I do not know of any Roman Catholic titles on the subject, except for Thomas Merton's translation of the Sayings of the Desert Fathers, which is incomplete; there are better versions available. These sayings should definitely be read.
Lastly, someone well acquainted with the Christian tradition of unceasing prayer might do well to read the writings of the medieval Yemenese Jewish scholar Maimonides, on prayer, which are interesting to the extent they seem to reflect and encapsulate the Patristic understanding, except Maimonides seems to have lost confidence in the idea of prayer as petition, which prayers are, but it is easy to see how someone outside of Christianity might give up on that aspect owing to the frustrations of being either continually ignored or tricked by demons.
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It is extremely important to discuss your rule of prayer woth your priest.