How do I tell what games were recently updated on Steam?

There's no way to tell from just the client or the SteamDB trick, since those aren't really helpful in cases where you don't download all the updates, but just some of them. Skip the first part if you just want to get to the answer itself.

Problems with the solutions of SteamDB and Steam

SteamDB will show every update of all the games you have. It also doesn't matter whether you downloaded the update or not. This list will probably have Dota2 somewhere in there, if you ever played it. So, this isn't very helpful. I don't want to see all that, just the stuff I downloaded... Relevant picture:

Again the Steam's solution is not perfect either, it's better since you can see 20 of the latest updates instead of 10, but if the game didn't publish the update as news, then it doesn't get into this page. Sure, I get a nice list of games but it's not nearly complete and there's stuff I don't want to see there, like the things I didn't download. Relevant picture:

Now, the only way I've found that works for me

I just navigate into the Steam's library folder. You can also right click a game and go to Properties -> Local Files -> Browse Local Files then go back one level to the common folder of Steam. If you have more than one library, you will need to look at each of them if you want to see all the games, but that's only to make sure you didn't miss anything. Your path will probably look like X:\SteamLibrary\steamapps\common for extra libraries or X:\path-to-steam\Steam\steamapps\common for the main library that Steam keeps inside its folder.

After you got into the folder where all your games are, you just need to sort by latest modified, and "Voila!", you have a whole list of updates that you downloaded in chronological order of the download date instead of the update's date (again, not perfect, but it works way better than the others):