Thanks.
Here's what I have found and achieved...
I have a Piwigo 15.0.0 install on a different CPanel host from the one where I had my old Gallery2 install (which will run mostly if I rewind the PHP version, but that breaks my other sites, e.g. WordPress, and is obviously risky).
On the Piwigo 15.0.0 I was able to install the Menalto2Piwigo plug-in. This is a PHP conversion of the old Perl script that people were talking about.
The plug-in has simple instructions saying where to put the photos and how to import from the database.
So, I transferred the old Gallery2 albums into the Piwigo directory it specified, and clicked Synchronise. Then I exported the Gallery2 MySQL database as SQL from the old server and created a new MySQL database on the new server, imported the SQL, and then entered the database details in the Menalto2Piwigo plug-in, and clicked the button. It also tells you to install the ExtendedDescription plug-in and then the captions appear.
It worked! I can see the captions - the user interface is differeent but I'll get used to it. The only issue I have found so far is that of the photos are on their side and need to be rotated, I think Exif problem for the early days of digital cameras. I believe there is a Piwigo plug-in that will do that.
I might still try the Gallery2 to Gallery3 conversion as suggested, and possibly try to install on the original server, but I am happy that I now have my old Gallery working on a maintained photo gallery package, and that I've still got the captions. The gallery is only for family members. I may now start to add to it again, and in particular I want to move albums out of Zenfolio, which I have been using but has got expensive and unreliable.