Ha! It! Really! Works!
I have to admit I was a bit sceptical of the outcome after I tweaked and added things and bits of my website at the Indiewebcamp in Nuremberg, setting up a service worker and offline caching things, adding a manifest file … while in theory I understand what all of this was supposed to do, I felt a bit dumb for not completly grasping how to control the stuff.
So as so often when learning new tricks on the web, it started with copy/pasting a working solution and trying to adapt this to my ideas.
But the one (visible) 'ahaa!' outcome -- getting the Chrome browser to display the 'add to home screen' prompt -- it hasn't happened yet. Even during my tests at the Open Device Lab Frankfurt, nothing happened. There is this thing; the Chrome browser kind of decides*) if you as a visitor have interacted and shown enough interest in the web site before it'll show this prompt. Apparently my one-time visits during the tests were not enough.
So yes, technically I now have my web site as a "progressive web app": In case you have visited some of my contents while being online, you then can browse and read the stuff while being offline. Other than that, I haven't dug deeper, so there's currently no storage of user input or interaction for use when being online again -- but all this is possible, though maybe not really useful for my website. I highly recomment Jake Archibald's talk on this year's Google I/O event, if you are interested in this PWA stuff.
(*)But also take good note of Jeremy Keith's article.)
For the moment I am excited that I got this to work, just for the sake of getting it to work. :-)