I have continued to research this question and here is the answer to the question that was so urgent to me several months ago: Orthodox prayer is indeed very different and in important ways from various forms of meditation, mantra, or even prayer in other non-Orthodox Christian sects. I this is one of the best explanations I've found and it's from Orthodox Christians:
Hierom. Adrian: One more question. Why do some people connect the Jesus Prayer to some other practices, for example, to the Hindu and
Buddhist mantras and meditation? Many people do not understand the
difference between those ascetic practices and the noetic Jesus
Prayer, the Christian prayer.
A. I. Osipov: If we turn our attention to the essential, then the
types of meditation you are talking about are reflections, internal
discussions. They do not carry with them the main condition for prayer
– repentance. Repentance is supplication. Supplication for what? For
our sinfulness, our inadequacy, our inability to live as the Gospel
commands. Prayer, as Bishop Ignatius writes, should be said with
attention, awe and heartfelt contrition. These things are not required
by meditation. Meditation, I repeat, is a concentrated reflection on a
great variety of subjects: theological, everyday, spiritual and moral,
There exists a very important and vital act in Christian practice –
the contemplation of God. However, this also differs from the
above-mentioned types of meditation. This contemplation of matters of
Christian faith and life goes hand in hand with humility, correct
prayer and reverent inward submission of our possible understanding of
any matter to God’s will.
This is the main thing that distinguishes prayer and contemplation of
God from meditation.
Now for the second thing. Turning to mantras, we enter the sphere of a
teaching that is decidedly, we could say, different from the Christian
or, more exactly, Orthodox teaching. Mantras, in some ways outwardly
resembling prayers or rather incantational prayers, are of a
completely different nature. They inherently imply belief in the
effectiveness of the very words pronounced, often regardless of the
understanding of their meaning. We see it in Hindu practice, for
example, in Japa mantra, which calls on people to repeat a god’s name
as much, as often and as quickly as possible, for the name itself
purifies man and brings him to the state of Samadhi. Mantras, if you
wish, are one of the elements of magic and are used in the rites of
pagan mystery religions.
A similar idea was promoted by the Russian name-worshipers. However,
it is not God’s Name in itself that sanctifies. The Name of God is
similar to an icon: it is a link to turn our prayers to the Archetype.
And human purification is accomplished not through the Name itself,
but through correct prayer with God’s name uttered in it, as the Holy
Fathers taught. When prayer is repeated mechanically, as many times
and as quickly as possible, then it “is not prayer at all. It is dead!
It is useless, harmful to the soul and insulting to God,” – as Bishop
Ignatius (Brianchaninov) wrote.
Currently too, we can see this tendency to understand prayer as a
mantra. Books are published which recommend saying the Jesus Prayer –
“Lord Jesus Christ, have mercy on me” – a huge number of times (14,400
prayers at one go!) from the very beginning. They recommend saying it
very, very quickly: 3,600 prayers per hour, that is, one prayer per
second (“his tongue, like a little engine, was repeating the short
Jesus Prayer non-stop”). This practice runs absolutely counter to the
Holy Fathers’ experience, which says that we are to say any prayer,
including the Jesus Prayer, without haste, paying attention to the
words of the prayer, with awe and a feeling of repentance.
Once you've spent a few months finding out where to look :) there are actually a large number of resources on this topic. I would highly recommend clicking through some of the other pages on oprelesti.ru as well as the writings of Sts. Ignatius (Brianchaninov) and Theophan the Recluse. St. Paisios the Athonite also has plenty to say on the subject. This article by Archimandrite Sophrony is good as well.
Finally, for anyone more interested in the topic, I can highly recommend the book The Gurus, the Young Man, and Elder Paisios. It is an excellent source of information learned the 'hard way' through geniune, spiritual experience. God bless you all!