Thank you all for your responses.
I have found that the situation is just what I had thought it was. You do NOT need a medical exam to get the Z Visa, but the college DOES need a medical certificate, in order for them to apply for the Foreign Expert Certificate, and to enable them to offer an invitation to you. And these are the documents you need to apply for a Z Visa.
When I last had to get a Z Visa, to re-enter China, I went to the Chinese Consulate in Perth, armed with the relevant documents from the college, but they still insisted on seeing the health report too, so I had to go back with it. I tried to explain that without it, I would not have had the invitation letter, etc., to show them, but it did no good.
Thankfully, every time since then, when I have left the country, I have had a current Resident Permit in my passport, so no need for a visa to re-enter China.
The Foreign Expert Certificate is what they work on, prior to our arrival, and this is needed, in order for us to apply for a Z Visa. The Resident Permit is processed once we are in the country, and is a follow up from the Z Visa in our passport.
Once again, many thanks.
Chinalin, Zhaoqing, Guangdong.