With help from Zend Support we have finally resolved this problem.
The simple answer appears to be that in Zend Studio 9 and higher the PHP file must be associated with a PHP Project in order for the "Report Problems As You Type" function to work. As soon as I associate a PHP file with a Project, the PHP editor reports syntax problems as I type.
I had never used the Project organization before within Zend Studio, but had simply organized each application within its own separate folder. So none of the PHP files here have been associated with a Project. I will need to change that practice in the future and always use the Project organization within Zend Studio so that the "Report Problems As You Type" feature will work.
Still, it is hard to understand why this should matter. If one is editing a PHP file within the PHP perspective, why should it matter that the file is associated with a Project? Clearly it did not matter in Zend Studio 8.0, but is a requirement in later versions.