Update trigger fired before insert trigger upon insert of new row
Solved: I have update statements for the same table inside the body of the INSERT trigger. Nested triggers are disabled on my database on SQL Server 2012 but after the restore on SQL Server 2016, the setting was somehow enabled.

Edit: It appears that the 'nested triggers' setting of the SQL Server instance was set to 'False' on SQL12 and to 'True' on SQL16.