There are many different kinds of meditation or prayer, Christian and non-Christian. However, I am confused regarding actual difference because these varieties, both Christian and non-Christian, seem largely to be arbitrarily constructed.
This question asks specifically about the rosary vs a mantra while I am interested in all methods generally. This one asks about sharing Bible references on the correct way to meditate but I think falls short as I do not believe the Bible is a meditation textbook. I also think none of the answers address my question as they seem to straw-man eastern meditation as harmful, demonic, or dulling.
That said, what are the qualitative differences between, for example, practicing the exercise outlined in The Cloud of Unknowing and counting your breaths? The author of The Cloud of Unknowing suggests repeating short, single-syllable words such as 'God' or 'sin' for the purpose of removing all thoughts and imaginings in order to be as receptive to God's grace as possible.
On the other hand, counting your breath seems to me to be an excellent way to remove all thoughts and images and, for the Christian, increase personal receptivity to the Lord's grace. Worth noting is that I am considering long-lived meditative traditions from mature non-Christian spiritual systems which have been providing benefit for at least a thousand years. I will not argue that the majority of new age spirituality is, at best, useless.
Is there a clear, defining line between meditations that are acceptable for Christians and those that aren't? If so, what is it? If I don't yet have the opportunity to learn from a Catholic priest, should I learn eastern meditation from a righteous man who believes it will lead Christians toward God? As far as I can tell, the answer is yes because it is the intention, correctness of practice, and unending goodness of our Lord that determines results.
However, I would much rather admit my own fault than attempt to outsmart two-thousand years of great saints. Answers should reference official statements from authority figures in the Catholic church. Thank you very much!