43inches, there's nothing in the regs you quote that would suggest it must be documented.
The first one talks about 'operational software' and a database is distinct from software, so that will mean the unit's actual operating software, not a database update.
The second one talks about 'quality control' of the update process and as such does not constitute a requirement to document updates.
I'm with the DR here, there's nothing in the regs or IR(EX) teaching that suggests database updates must be documented. They are obvious in that they're in front of you when you switch on the receiver.
Sure there may be companies who prescribe documentation as part of company regulations to make the process of updating and checking more transparent internally, but that doesn't say we must all do it!