OK day sleep sounds ok so doesn't look like overtiredness is an issue, more so that she is having a bit too much on the day sleep... at 12mo 3 hours would be maximum to ensure good night sleeps & if she is falling asleep without a bottle (does she have a paci/dummy? if yes that may be part of the problem) so probably doesn't have a "sleep prop"
I think given her age one of the issues is Separation Anxiety & if you are working she may wake at night to get some of her "mummy quota"... I know that was an issue with my ds1 & I know others have the issue too.
I will be honest here I think letting a child cry to sleep isn't fair to a child (this is my opinion) & I can't (in my mind) discount that she isn't sleeping well due to the emotional issues of not feeling her needs are being met... (there are people - professionals- who may dissagree with me on that, but - Tracy Hogg author of the Baby Whisperer Books was a firm believer it wasn't good to let babies Cry it out (CIO). I certainly firmly believe babies cry for a reason & while we may not see that reason as important to us, it is to them (who are we to judge others needs) & by ignoring their needs I think we send a message that says they aren't important as a person (think about when an adult ignores you... how do you feel) .... further to that, she may have worked out that you don't respond to her crying at the start of the night so no use in doing that, but if you do respond in the middle of the night it gets a reaction & she gets attention so she feels "validated" by you, so she knows that at least in the dark of the night Mummy cares enough to answer my call.
My suggestions may go against your ideals, but I think you have to give her as much attention as you can & build her trust at times other than in the middle of the night... you still need to respond to her in the night. The water may also have become a prop that she uses to get back to sleep at that time, so maybe try just comforting her with words.
Also I have found it is important to give them a little while to see if they settle as sometimes night crying is just part of their sleep... I normally count to 50 & evaluate & either keep counting if the crying is settling or start to get out of bed... I made the mistake of "rushing in" to every sound my ds1 made & I think I made the problem worse
HTH & please don't feel I am judging you for letting her cry, I know of people who have had no problems with doing it & their babies sleep really well, but you have had problems & in my mind I can't discount it as a reason given nothing else sticks out