In Texas a motorcycle has:
engine between legs
does not have a steering wheel.
A morgan is not a motorcycle here.
Actually Poppa - it is a motorcycle in Texas. MY M3W is registered as a motorcycle, and I needed to get an M endorsement on my license to drive it. The definition you quote is on the DPS website...but when I called they said they consider the M3W to be a Motorcycle/Trike.
With that said, Texas does have a special license endorsement for three wheelers, and you need to take the class on a Can Am Spyder - type bike, but that is another option for licensure in Texas.