Don't worry, everyone calls them TGI's. It just always bugged me.
Sounds like you have built it right. The 2x4 strapping is the best thing you could have done. Not only does it offer strength, but it will give you air flow if you insulate your ceiling (which is probably a better way to spend your money than even the overhead heater).
Did you put a chamfered ledger on your top cap plate? Cutting birdsmouths into the joist normally isn't a good idea.
What about your overhang, do you scab on 2x4's and extend them out?
not that any of that really matters.