This 'diminished run' is one of the most prevalent cliches in popular music, from Ellington to Sondheim. It's all about the voice leading. Sometimes we label the intermediary notes with chord names, sometimes we don't bother and just write the notes. It only really works with the one specific voicing - a run in parallel thirds. Filling in 'full chords' isn't necessary and is probably sabotage.
Here it is with chord symbols and as notation. Recognise the pattern - as I said you'll come across it a LOT. And see what you're required to play when you find that chord sequence. Just that simple run of thirds.
It's related to this other cliche, the 'Basie ending'. Again, you COULD confuse it with chord symbols... But you probaby shouldn't. Some things just need to be specific. Symbols don't cut it.