I can now reproduce the issue reliably!

Thank goodness!
I was beginning to think I was going mad :)

Thank you for investigating this with so much energy and perseverance.

Perl 5.16.3 is now over 8 years old and no longer supported. You may want to ask your webhost to upgrade their Perl.

I have asked before and the answer has always been no...
If I want a later version of Perl they say, I need to change from shared hosting to either VPS or a dedicated server.

My yearly shared webhosting contract ends at the start of November so I guess now is about the time to make the overdue switch to VPS.

My concerns about VPS are that I don't have the time to manage it if (when) anything goes wrong plus the very real possibility that I could break something essential. With shared hosting, the inner workings of the webserver are shielded from me and I am just left to deal with my own code. With a VPS or dedicated server, much more could go wrong!

Perhaps I should buy a cheapish refurbed PC and install some flavour of Linux on it. That way I get to find my way around a bit more before making the switch to VPS. It would also give me something to test things on before touching the live webserver...