Well, it seems I have blundered about and by accident found a solution.
Although neither my main iMac computer, nor my MacBook laptop could connect to the Plex media server, both continually getting the message “Connection refused” I discovered that the Plex app on my phone could apparently see the server from a previous connection, though it wouldn’t work in a browser on the phone.
The Plex app said it could see my plexserver at a plex.xxxxx.dyndns.org web address. And while the Home page opened to a list of recent films, the media sources listed on the left only showed Music, Photos, and Music from my server.
I could play music from it, but couldn’t see Films.
So I thought I’d try what my iPad could ‘see’ of the plexserver on the local network. Again, browsing to the ip address and either port 32400 or 32401 didn’t work. But when I started the Plex app, I got for the first and only time a popup inviting me to Claim Server. I did that.
It also showed me a choice of TWO plexservers, each at the same URL of the form plex.xxxxx.dyndns.org. So I picked the one shown as default. Then they both disappeared, and all I saw in the dropdown instead was Sources.
The Home page on the iPad this time showed only online sources including Plex Movies and TV, web, News, podcasts, etc. But I was able (again more or less by accident - I can’t find it again) options to add a folder each for my different media types for Film, TV shows, Music, and Photos - (already preselected as /var/flexshare/shares/plexmedia then subfolders …/video/movies, …video/TVshows, …/music, and …/photos).
I was then able to edit my Home screen and pin them, as well as unpinning some of the other online sources I don’t use or need.
Went back to laptop, and tried again using Chrome, after clearing the browser cache. Still didn’t work. But Safari did, eventually, after warning that my connection although https was not secure, as the Certificate is untrusted, and if I REALLY wanted to connect I needed to give my computer password.
And then it started to work again.
After confirming those folders and pinning them, all seems well again in Safari.
Then I repeated this sequence on my main iMac in Safari, and did it again in Chrome, after clearing the cache.
Now all seems well. I still have absolutely NO idea what set off this whole series of problems, other than (perhaps) a weeks-old series of updates from media server v1.17x to 1.18x, then 1.19.
It hadn’t worked since the first upgrade to 1.18.
But now it is working, on 1.19, installed manually a week or 10 days ago. HUGE relief.