Starting from the standpoint that your peers or reports are lying to you is never going to work. Treating people like kids means you'll get kid level responses.

If someone is lying, it will eventually come out. That fake report one day will turn into a week and eventually they will get fired. But, don't kill your entire team on the off chance that someone is trying to game the system.

Assuming that you really think that the members of your team are lying to you like this, it's easy enough to validate.

The first one says they're writing a spec. Well, three days later, where's that spec? Next one is producing some designs, so same. Sure someone saying "$TOOL/$NEW_FUNCTION/$REPLACE_FRAMEWORK just didn't work out", could be cover for "I was playing Pac-man all day", but if you ask for details, BS should become clear pretty quickly.

And again, that's starting from the position that you think your teammates are probably slackers who would rather lie than work. Which is not really a great position to hold, IMO.

It doesn't really matter which.

Look, if I'm leading a group like this, I'm going to have an idea of what your output is like in an average sense, both compared to your historical output and the group as a whole. If you blow off a day it's not really going to matter. (although I'd rather you tell me you needed to blow off a day than try and hide it, sure).

However, if you blow of enough days we're going to have a conversation about how your performance is dropping, and why, and discuss your ideas about how to turn that around. If it doesn't, we'll have another conversation, and so on. Eventually if none of this works we'll have a different conversation (about why you are leaving).

This doesn't actually have anything to do with reviewing daily/weekly reports. Those are for different things.