Often they are just not used, though.
Probably contrary to the Japanese custom as described in the article I personally always feel slightly self-conscious when using them (to hand someone the money). It feels distant and maybe a bit impolite to me, though I don’t know how shared that feeling is. (Please note that I’m not claiming that using the tray is somehow inherently impolite or distant. When it comes to etiquette the concept of universal truth is even more problematic than it otherwise already is.)
My usual approach is to hand over the money directly if there is little time and the cashier is already looking at me, expecting the money, and to put it on the tray if the cashier is currently still doing something else and I have a bit of time to properly prepare the money while the cashier is looking elsewhere. (It basically gives me a proper place to put the money when the cashier cannot yet directly take it.) I’m not sure how shared my feeling and my approach on this are among others.
Also, in my experience cashiers differ wildly in how they will use those trays to hand you your change. Some will use them, some just won’t. I never payed any attention as to whether there’s any pattern to that, though. I mean, is the proper use of that tray part of the instructions when starting somewhere? Is it just personal preference?