In reply to CPH:
I would agree with that interpretation.
The requirements within Europe depends how long you have been there, not your nationality. So eg if you had been in France for 2wks the fact you originally travelled from the UK is irrelevant.
There are also different rules if you are transiting a country, and these can be found by using the same website and using the transit function eg start UK (rest of world) add France as transit and destination Italy and vice versa.