Here is the Idaho "motor vehicle" statue. The problem is, there is no Idaho definition of self propelled.
One of the many definition of self propelled is "Moving or able to move without external propulsion or agency:" (Merriam-Webster definition). Can a motorized bike fall under that definition? I would say it's possible.
I would also point out that in the moped definition, it specifically says it is not titled. That lead me to believe that the last half of the definition really excluded mopeds, but ITD disagrees.
(g) Motor vehicle. Every vehicle which is self-propelled, and for the purpose of titling and registration meets federal motor vehicle safety standards as defined in section 49-107, Idaho Code. Motor vehicle does not include vehicles moved solely by human power, electric personal assistive mobility devices and motorized wheelchairs or other such vehicles that are specifically exempt from titling or registration requirements under title 49, Idaho Code.