Is it true that domains registered before 2004 have a totally different way of getting PageRank? Like 'Pre-2004' domains are highly desirable because they get PageRank based on old easier criteria.

No, it is completely false. There is obviously no difference between 2004 domains, 2005 domains, 2006 domains. All the domains get reputation in the same way. 

Literally, there is no extra value in buying a pre-2004 domain or pre-Google IPO domain or whatever you want to name it. There's no difference absolutely. 

Just ensure to get a domain which  works well for you. There is no need to think about if it was created before 2004.