Is there a way to set up Plesk so that it gives more detailed access logs? Like entrance/exit pages, what pages users were on, browser type, search terms, and so forth?
Sounds like you're looking for Domain-Statistics. Check for each hosting configuration of the domain. There should be the Webstats enabled. Then you can look at the stats in the Browser for each domain.
Currently we have it set up where urchin is reading the access log files. So the plan is to direct urchin to process those ones instead of /var/www/vhosts/*/statistics/logs/access_log
Because that one doesn't have all the information we want.