Your trip does not constitute an international itinerary. It is one international itinerary and two domestic itineraries, unless all the flights are booked on the same ticket.

If all of the flights were on One World carriers, it is possible that you would be able to check your bags all the way through for your return, i.e. Intl-NYC-RNO-DTW; although with three different itineraries that may be a push (it definitely works fot two adjacent OW itineraries on separate tickets). However, since Horizon isn't OW (even though it's an AA partner), I'm pretty sure the LAN agent won't be able to check you all the way to DTW. Of course you could ask as YMMV, but I wouldn't get your hopes up.