In a great deal of contemporary English writing, the two terms are used synonymously -- the writer doesn't necessarily distinguish an apparent difference between the two terms. Even dictionary definitions, as you noted, overlap considerably. However, correctly or incorrectly, I believe that there is a difference between the two terms, but that distinction doesn't lie on the fault line you propose. As an alternative, please consider the following example:

In the 1860's southern U.S. states, slaves were given their freedom (no longer bound to servitude). With that 'right' restored, some former slaves took advantage of their new liberty and headed West to live as they chose.

In this context, I consider 'freedom' to be more fundamental and should be considered a 'right', however it was controlled by humans and imposed on others. 'Liberty' in this case required freedom as an essential prerequisite, but could have been limited by things there not inflicted by others, such as the physical condition and age of the West-bound former slave.

I don't know if this distinction is shared by others, but it is one that I have adopted.

Regards,

Phil