Basically, can I leave a handgun locked up in my trunk and drive around wherever I want?
You've always been able to do this.
The new law says that as long as the handgun is concealed
you can have it in the passenger compartment and readily accessible to you. There's nothing in the law that suggests you can't have it in a concealed holster, so you could even wear it as long as it's concealed.
Common sense would suggest that there needs to be a lawful means of transporting a firearm with a motorcycle.
Trying to interpet the law based on the assumption that it must conform to common sense can get you into a lot of trouble.
For example, a person carrying in a car WITHOUT a CHL can drive into a 30-06 marked parking lot legally while a person carrying in a car WITH a CHL can not. Where's the common sense there?
I don't know exactly how motorcycles are treated, from a legal standpoint. That said, motorcycle carry with a backpack seems like it could be legally problematic. ASSUMING that it's legal to treat a motorcycle exactly like other vehicles even though it doesn't have an enclosed cabin, then you could carry in a compartment in the motorcycle or concealed on your person, or in a backpack while you're actually on (in?) the motorcycle.
HOWEVER, if you don't have a permit and you get off (out of?) the motorcycle (other than to walk up the drive to your house or to some other place you're legally able to carry concealed--i.e. private property where you can legally possess a concealed firearm) then you're breaking the law. So if you have to stop for gas, take a restroom break, get something to eat, etc., you would have to leave the backpack on (in?) the motorcycle to remain legal. Without a CHL, your best best would probably be to carry it in a compartment on the motorcycle.