Sgisela Glad that your hand is ok, but I think the consensus medical advice is not to use ice water but cool running water (for 20 to 30 minutes).
It's been a long time, but I recall reading a study on reversing burn-damage using dry ice(!!). The images were fascinating, and the key take-away was that there is a lot of damage that continues post-burn if you can't conduct the excess heat away from the internal tissues. Conversely, if you can conduct the heat away before the cell walls burst, then there's surprisingly little damage. But, you obviously don't want people applying dry ice to themselves, or pressing blistered skin up against blocks of ice, or doing anything for so long that they compromise blood flow for too long. So, the advice back then was - assuming no broken skin, and a small burn - to apply an ice cube (normal ice, not dry ice) directly to the burn, but for no more than 15 seconds, remove the ice, let the skin warm slightly, then reapply the ice, repeating the process for 20-30 min. I think the preference was actually for a circulating ice bath, like the sort athletic trainers often have, but the authors acknowledged that people don't usually have one on hand.
FWIW, I've used this technique (ice on, ice off & warm slightly, repeat) on my own (thankfully small, never terribly serious, knock wood) burns for the past ~ 20 years, with great results. And when I get lazy (i.e., I feel like it's "good enough" after 5 min), I can tell the next day. But I had a lot of experience with circulating ice baths in college, so it may be that I have a better calibrated sense of when to know that I'm getting close to 'too much cold'.
I've been told by many doctors that broadly-disseminated medical advice in the U.S. is usually made under the assumption that the reader won't read carefully and is likely to think that if a little of something is good, then a lot more must be better, so it's considerably under-dosed, so to speak. Especially once the editorial and legal departments get involved.
ShiroKuro, I'm glad your hand is okay!