How to restore a JR site with the Database backup file only?
Installation and Configuration
The problem here is that your 5.3 backup likely has modules "active" in it that are NOT installed on your system. You need to look in the "jr_jrcore_module" table and make sure you have installed EACH module that is in that table - if you're missing even one then you'll get that error, because when JR "starts" it is going to run an include for each module that is active in that table (as defined by the module_active column being set to "1").
Jamroom 5.3 is fully upgradable to 6.1 no problem, so that should not be an issue. The repair.php will fix it all up, but you have to have ALL the modules installed on the file system that are defined in the jr_jrcore_module table.
I see you're not on our hosting - have you considered checking out our hosting? If interested let me know - we can get this working for you.
Let me know if that helps.