Besides actually giving your character a move action, what other ways can someone be moved(not placed!!!)?

For example:

Is someone considered moved if they were carried?

Is someone considered moved if they were mind controled and given a move action, or power action where they actually moved from one square to another?

What other ways can someone be moved?

Carried characters are placed.

Any power or ability that says Move or Movement is actual movement, no matter what type of action it is, even Mind Control.

Also, although in the description of Knock Back it says movement, the player's guide says it is not.

Not just the PG. From page 10 of the RB:

Game effects that activate as a result of a character moving are not activated by a character moving due to knock back.

Didn't even notice that it made it into the Rulebook.

i am pretty sure Breaking away is a movement as well. why are you asking if i may ask that?

Breaking away can be done when a characte moves, but isn't itself anything other than, well, break away.

I am looking for regular or special powers that allow you to move a character or cause a character to move.

Then look for any powers that say move. If they say place, then it won't work.

Open the PG in Adobe Acrobat Reader, hit "ctrl-f" to search, and search for move.

I am asking here in case I missed something, or there was a wording change that I missed or if a character has a special power that would apply that's not in the PAC.

Well, there are no special powers in the PAC... so from that POV, yeah, you've missed a bunch. I highly recommend using Ctrl+F in the Player's Guide as already suggested. If you don't know where to find the PG, click the 'rules' link in my signature.

Moved and placed are different terms and not interchangeable.

Actions in which characters move or can be moved:

Move Action (in itself)
Leap/Climb
Phasing/Teleport

Power Action (move + attack)
Charge
Hypersonic Speed
Running Shot

Alternatively, Mind Control can be used to give a character a move action.

Characters do not move when being carried, knocked back, or placed by TK or a special power/trait/effect.

You could give a character a move action and not move it (0 squares), but because the character did not leave the square it occupied when it was given an action, it was not moved. Zero is zero.

If your question involves Deathstroke's Precisely-Timed Strike SP, then a character that is given any of the listed Move or Power actions would trigger the SP when they become adjacent.

This is incorrect. In this game a move of zero is movement.