I'm sure most non-mouse-over healers have noticed this even if they didn't know why. You click on a raid frame to switch targets and cast a spell, but your spell ends up going to your previous target. You know I came across this very same issue back when I started using the SnowFallKeyPress add-on which caused my hot keys to cast on button down instead of button up. I just figured the change made it faster than the target switching and got used to it over time. Then SnowFallKeyPress broke and I never bothered to fix it and eventually Blizzard made it's functionality part of the game and I was back to the same issue.
Turns out the root of the problem is that clicking a raid frame doesn't actually change targets till you let go of the mouse button. So when your spell cast happened when you released the key and the target flipped when you released you mouse button the two events always happened in the right order. Change the ability to cast as soon as the button is pressed and the targeting is going to lag behind.
Once I realized what the problem was I started looking for solutions. The healing specific add-ons like Healbot and Vuhdo had apparently figured this out way before me and already had the ability to target on mouse down. Pitbull however did not much to my surprise and I spent a little bit of time looking through the LUA code to rectify that without getting anywhere. Then I remembered another add on I'd used before, Clique. I downloaded it and installed it to find that the very first checkbox in the options did exactly what I wanted.
I suspect that Blizzard will notice the benefits of targeting on mouse down in time, but in the meantime I've got a solution in hand before I got used to hitting my action buttons later to compensate again. I can't believe I didn't catch this the first time I went through this, but I suppose I wasn't healing as much at the time and was mostly impressed by the faster reaction time when DPSing, interrupting in particular.