That is all correct: you have to fulfill the most restrictive of all applicable regulations.
Example: you may be allowed to fly a SE with retractable gear etc. (= 'complex aircraft' and with > 200hp: 'high-performance aircraft') as PIC with your European license in your country, but since FARs require the above mentioned checkouts, which are not known elsewhere, you still have to have them as well. In case anything goes wrong, the insurances sure as hell know exactly all the regs that are required!
In your case: flying with the applicable European license within the country that issued your license is fine. But stay within your airspace or have an FAA license.