AOL didn't comply until the judge signed the injunction order. AOL posted a notice about it at the time. which was, IIRC, 1997 or 1998. Right about the time LUG was securing the license. Pre DMCA. Once the injunction issued, it's not threat of suit, it's invoking the weight of the US government.
In the US, the injunction requires convincing a federal judge that (1) you've made reasonable attempts via C&D letter, (2) it's more likely than not that it's an unauthorized use of copyrights and/or trademarks, (3) that it's likely not fair use, and (4) they haven't yet complied with the C&D.
Paramount got a judge to agree it was more likely than not that an an authorized use of their IP was happening in a fixed medium (you could access the prior posts) and appeared to violate their IP. He also agreed that it likely wasn't a fair use.