Since most of you guys who post here are gamblers, may I ask you: how many of you take screenshots of their deposits on casinos?? OP did?
Speaking generally i never took screenshot of my deposit in the gambling casino. Also why should be take screenshot when everything is visible in the deposit history. Still if you do not trust the gambling site and think they can manipulate with the deposits, the blockchain data will reveal everything.
Also, these days it is very easy to manipulate and tamper / edit the screenshot to produce a fake one, so i think that transactions on the blockchain are the safest way to provide the proof of deposits.