For instance your "three landings in the last 90 days" is not specifically restricted to a license/country/aircraft registration. (Although it might be restricted to an airplane class/type - I'm not quite sure about that one.)
Its in the type or class ie 3 landing in an Airbus doesn't do anything for your SEP currency.
I wonder if that is a bad thing
I have had to "clean" 3 peoples log books of CFI's opinions on the pilot. If they had been left in they wouldn't have had a hope in hell of getting a job when a CP saw them.
All three of them now have commercial jobs in fact one of them is up for a LHS on a 777 in the very near future.
Its not the case of running two log books its just that you run a FAA one and transfer the hours into your main one at a later date. I know 2 people who have lost there log books and its something I never want to go through myself