just a follow up to let anyone who strays here know the outcome

I decided that two things were going wrong on my server. SSL certificates were not being handled properly and the two separate paths from the server root to HTTP and HTTPS folders was not working properly.

I have ignored the first fault for now because I think it is a red herring from a Zen Cart point of view.

I deleted my domain and re-configured it so that HTTP and HTTPS are served from the same root folder. I have a VPS, which makes doing this easy. Having separate folders for the two protocols was the default setting when I set up the domain originally.

I uploaded my site, set up the database and installed Zen Cart. It now works properly. I can access admin on HTTPS but using the root SSL certificate, which I will fix later, hopefully.