I like this:
> Sometimes liberty is differentiated from freedom by using the word "freedom" primarily, if not exclusively, to mean the ability to do as one wills and what one has the power to do; and using the word "liberty" to mean the absence of arbitrary restraints, taking into account the rights of all involved
It's from Wikipedia, and it implies this is the modern take of the definition. I think it's how I think of it as well. So it is neither of the two you mentioned, but a combination of them with the focus being the balance between them.
Liberty would assume all have rights they are entitled too, and that none shall arbitrarily restrict ones ability to do as they please, where non-arbitrary is defined as not restricting of other's rights.
I don't think it really puts people against each other. Some people simply disagree with liberty and favor freedom instead. Which would mean, some people want to be free to do whatever their power allows them too. You can think of it as whatever I can get away with because I'm more powerful. It would mean if I'm stronger I can strongman my way into doing more things, same if I'm richer, more influence, etc.
Fundamentally it's a disagreement with your objective. If you don't accept that the less powerful still deserve certain rights, or that power should not dictate rights and restraints, there's no amount of discourse to be had, you will be optimizing for different outcomes.
I also find the framing of rights as someone else's problem misleading. It is not someone else's problem, oftentimes it is because of restraints society imposes, the other person's problem is due to their restraint on other people's rights. For example, that I can't just walk in your house and sleep in your empty bedrooms as I please, and eat the food sitting idle in your fridge, or build myself a cabin using wood from your trees and on your land, those are all restraints society is imposing on me. So if I'm now homeless and without a job, I cannot just do these things to provide for myself shelter and food. But if you believe everyone has the right to shelter and food, and you are restraining my ability to get them as such, you need to offer an alternative, it isn't entitlement, it's the trade for accepting the restraints being pushed on me.
For me, it's the fundamental agreement, you accept the restraints from laws in exchange for rights. If the rights don't come, you're not getting your side of the deal. Now off course people can impose restraints with power instead, and that's almost always what used to happen and still to a large extent does today, but at least we seem to try harder today to be just.