Wow-
Looks like you got a lot of advice, so please excuse me if I repeat (I didn't read them all).
I too have a busy, busy daughter (I say she's my spunky little monkey). She also hit her "terrible two's" at 18mo. She is now 3 and getting better, so be patient... you'll get there. Keep re-enforcing the positive such as that's an owie - be nice, gentle hands - no hitting, etc... (you are already doing).
Some may say she's too little for a time out, but with my daughter she understood soon enough. I always told her to say sorry to someone she may have been unkind to or hit... have her hug them and explain why it is wrong. If she chooses not to, then she sat facing the wall until she could (it was cute how she said "saawee"). If it was something she did and not to someone, she needed to say sorry to me. Over time, she has little time outs and knows when she does wrong where she apologizes without my telling her.
Also, counting to three helps. I tell her she needs to __(change what she's doing wrong)_ when I get to three, or she will get a time out. She'll get the knack of it after a few time outs... she'll then wait until 3 in the beginning, then maybe start listening at 2, then change right away at the start of 1. It's somewhat funny how my daughter will now say - when I start at 1.. "I'm coming!".
In fact, to make you feel better, when my daughter was starting this.. she hit my husband and I scolded her explaining that she needs to be nice to daddy and hitting is not aloud. I said that it hurt daddy and she should say sorry to him... she chose to take a time out (when I didn't even tell her) instead of saying sorry! That is just one of the stories where she gave herself a time out. Kids understand more than we give them credit for.
Just be consistent in what you decide how to handle this. She'll understand when you keep following through with your choice. I always make sure she understands why it was wrong and what is the better choice (keep it age appropriate - simpler now, more explaining as she's older - such as "be nice right?" to "that's not nice to... will you be nice now?" to "what did you do?"... will you...now?").
Many times I've felt like pulling my hair out with my daughter, but I wouldn't trade her for the world!
Good luck and I wish you the best (and all the patience in the world). I feel for ya - I'm sure we could share many similar stories. In time, it will get better. I think it took all of the two's with my daughter and she still tests me. My husband and I joke that if she would have been the first (I have an opposite first born - 8yr old son), we probably wouldn't have had a second :)
~SR
P.S. -- Always praise the positive/when she does something good. She will get praise for the good, and consequence for the wrong, and soon she will choose good things over the bad because she'll like getting praised. :)