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You should be careful putting any faith in some screenshots that you might have saved, because they can be so easily manipulated and faked these days that no support team is going to put that much trust in them. If you're in an argument with the casino itself over what has been said in a previous support chat log, it's a toss up on what they'll do with that info.
I don't see it as such a big deal. A conversation with support (usually) always ends with a support ticket with a unique ID that you can always refer to - if you have a screenshot of this ID. Some exchanges/casinos also send you the complete log of a conversation/ticket by email after it has been completed.
A casino would therefore have to actively refute its own support staff's statements or claim that it no longer has access to the tickets. Even then, they would have to prove that you faked the screenshot. This would require a considerable amount of deception on the part of the casino, which would never hold up in court (customer communications must be stored for a certain period of time, usually several years).