Is Centering Prayer "Transcendental Meditation" or another form of Eastern meditation?

No!

TM asks people to empty their minds, to empty themselves.
CENTERING PRAYER asks people to quiet their minds so as to be filled with their awareness of God. Centering Prayer is a lovely way to choose to turn more completely toward God, to increase intimacy with him. It is a way to more fully acknowledge--and allow him to strengthen our awareness of--our already-existing and subordinate connection to him.

TM asks people to go deep within themselves. ("Imagine that you're on an elevator going down, down, down....")
CENTERING PRAYER does not ask you to "go within yourself"; going within yourself requires *you* to be in control! It does not ask you to bury yourself in yourself; it helps you to bury yourself in God! CP helps us to strengthen our spiritual connection to God, not the self, by disciplining the mind and body. CP helps us to sense God's presence more completely, to increase our spiritual awareness of him throughout our day, thereby giving more and more control to HIM!

TM, similar to Hindu meditation, asks people to essentially hypnotize themselves, merging with the universe of which "god" is a part.
CENTERING PRAYER asks people to quiet the mind and the will so as to hear God and be "in the Spirit" more completely, to rest in him. The relaxation that often accompanies centering prayer is spiritually restful and is NOT the type of self-hypnosis that leaves a person spiritually vulnerable. (I know of many ways to relax that aren't evil! In fact, God wants us to learn to rest in hm!) Transcendental meditation's goal is for a person to enter a zen-like state and actively attempt to "merge with the universe". But CP does not have you focus on the relaxation--that would have you take your eyes off your purpose: to focus on being with God. Any associated sensory experiences are just side effects of this rest, not the goal.

TM leads people to believe that they can be merged with "god" or become like gods.
CENTERING PRAYER acknowledges that God is separate from us, the Creator of all. Centering Prayer doesn't merge us with God; it strengthens our awareness of our already-existing, subordinate, spiritual connection to him.

TM and similar types of meditation encourage people to believe that since "god" (part of the universe) is in us, then we can reach out (pray) to god IN us.
CENTERING PRAYER acknowledges that although God dwells within us, He cannot be contained there. Centering Prayer has us reach out and connect to God from within the depths of our selves.

CENTERING PRAYER does not have people "capture" God within themselves. That's not possible. But if you believe we were made in his image and are souls merged with flesh, then we have a spiritual connection to God who is separate from us. However, our busy minds get in the way and the incessant demands of physical life interfere with the spirit!

CENTERING PRAYER should always keep you focused upon God as Creator, God as Love, God as our strength and our shield. Just as fasting allows us to strengthen our will and decrease our body's control over the mind, Centering Prayer gives us tools to stop the swirling and experience the fullness of God's nearness, love, and strength.