above it correct. An observation - i've traveled plenty under either US or non-US names (same situation as OP). I've generally found that using my US name on ticket and showing my US passport first gave me better treatment and less scrutiny that going with my non-US foot forward (even if i eventually I show both passports). Everything from airline ticket agents to TSA to gate agents and even foreign immigration.
I can count almost 100% that if my ticket is in my non-US name, I will have "random" secondary with very thorough examination of my carry on. With US name, rarely happens. And just less questioning by everyone overall.
on a more practical level - all my FF accounts are in my US name and if I'm getting miles for a flight, I book it in the same name as my FF account. maybe this doesn't apply to the OP.