I see liberty and freedom as the following, without doing any research at all except your video:
Liberty is the right to do something (legal protection)
Freedom is the opportunity to do something
Free will is the ability to choose and perform something
An example might be driving a car.
- When you turn 16, you're afforded the Liberty to drive unrestricted - without fear of punishment. (Given assumptions/restrictions such as driving in an emissions and electrically safe approved vehicle, following the laws of gravity and not flying in the clouds, and abiding by the DVLA)
- Freedom is having the resources to actually drive, such as money, time, a parking space, a licence etc. This is partly 'driven' (pun intended) by equity.
- Free will builds ontop of that and is basically a decision tree of to drive/to not drive/to buy a car/to not but a car/to run people over/to clean your car/to pick up your nan from hairdressers, and all possible permutations.
Positive and negative liberty as you described them seem to map to my definitions above - negative is my liberty, and your positive is my freedom or even free will. A parasympathetic pathway/prohibition/hinderance/negative liberty is a bouncer stopping you from entering a club. A sympathetic pathway/entitlement/positive liberty is you having the personal ability to choose to go to the club in first place, the money to pay for entry, clothes on your back, etc. You can see how free will gets mixed up there.
In the case of the driver's addiction, in my opinion their free will is being obscured. Legally they can smoke. They obviously have the money/freedom to buy cigarettes. But the addiction in their mind is simply cutting off options from the decision tree. It closes doors and narrows the possible permutation paths down to forced destiny.