In (4), though, I do not get the French exception. There are many countries in Europe with a president and a prime minister
In most of those countries, the role of the president is mostly ceremonial and/or has to do with foreign policy outside of EU.
Germany and Italy are exceptions, not the rule
22 countries: Prime minister
5 countries: President
https://en.wikipedia.org/wiki/European_Council#Members
Yes, but what does it have to do with the original point about countries with real power presidents?
In (4), though, I do not get the French exception. There are many countries in Europe with a president and a prime minister