So I've fixed this issue myself. I noticed that in my new database, the ks7hv_update_categories table was missing, and thus I couldn't open the ks7hv_updates even though it appears to be there in Phpmyadmin.
I've restored my last backup of the Joomla 2.5 version the site and I exported the ks7hv_update_categories and ks7hv_updates tables.
Opened up the exported file and copied the queries from there.
After that, I opened the SQL tab from the database of the Joomla 3 version and first I executed: DROP TABLE ks7hv_updates;
Then I pasted the query that I copied before:
CREATE TABLE IF NOT EXISTS `ks7hv_update_categories` (
`categoryid` int(11) NOT NULL AUTO_INCREMENT,
`name` varchar(20) DEFAULT '',
`description` text NOT NULL,
`parent` int(11) DEFAULT '0',
`updatesite` int(11) DEFAULT '0',
PRIMARY KEY (`categoryid`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 COMMENT='Update Categories' AUTO_INCREMENT=1 ;
CREATE TABLE IF NOT EXISTS `ks7hv_updates` (
`update_id` int(11) NOT NULL AUTO_INCREMENT,
`update_site_id` int(11) DEFAULT '0',
`extension_id` int(11) DEFAULT '0',
`categoryid` int(11) DEFAULT '0',
`name` varchar(100) DEFAULT '',
`description` text NOT NULL,
`element` varchar(100) DEFAULT '',
`type` varchar(20) DEFAULT '',
`folder` varchar(20) DEFAULT '',
`client_id` tinyint(3) DEFAULT '0',
`version` varchar(10) DEFAULT '',
`data` text NOT NULL,
`detailsurl` text NOT NULL,
`infourl` text NOT NULL,
PRIMARY KEY (`update_id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 COMMENT='Available Updates' AUTO_INCREMENT=121 ;
And after that I inserted the values (used the query from the same exported file) and after that it worked fine again.