THANK YOU THANK YOU THANK YOU
I believe it is now fixed.
I am indeed running Plesk. The provider had all the various versions preloaded, but only made the lowest one the one used.
Different versions of Plesk will vary, but there are two separate places, each labelled "PHP settings".
One invites a person to physically install different versions of PHP.
The other invites the person to select from those currently loaded.
Althugh version 7.0 is available, I selected from the highest of the 5's, since you had mentioned that we needed only to exceed 5.5, I think. I installed 6. At first it wouldn't work, and then suddenly it did.
I am so happy.
Thank you for bearing with me.
I am going to go off and do some more testing to be certain, but I think it is done.