It is a good idea to print or download evidence from the Wayback Machine or other public archives, in case it gets scrubbed.
A website is just a publicly viewable computer file, stored on a computer. You can fake file dates just as you can with files on a home computer.
You can check for anachronisms. For example, a file in Word 2010 format with a 1999 file date, or an “old” file that contains a new font.
There can be metadata in a file that the user is not aware of. And that could give them away.
Depending on the website being faked, there could be a lot of backend software that could introduce clues (database files, PHP versions, etc.)
Right now, I have saved (print & digital) the website info from the way back machine. My opponent will claim that their own archiving system is correct and the way back machine is wrong or not admissible. The info on the website would have been there for a long time, at least 6 months, so way back must have sampled it.