TexasCHLforum.com

But aren't "dirt bikes" [motorcycles] considered motor vehicles?

I bet only if they're registered with the Texas Department of Motor Vehicles.

I don't know for sure, but I would bet not. The statue says that the exception is for a person "inside of or directly en route to a motor vehicle
that is owned by the person or under the person ’s control."

Is it even legal to ride an ATV on a public road?