They are not exactly interchangeable, but the distinction is very subtle. To illustrate, let me first change your example sentences into the forms I find most agreeable.
The people were free of the barbaric dictator.
The mashed potatoes were free of lumps.
I wish I could be rid of this terrible cold.
This demonstrates that "free of" is most comfortable for me when used to indicate that something no longer is beset by an entity that had been pervasively enmeshed in its very existence, as a dictator controls every facet of a people's lives, as the lumps in mashed potatoes influence every bite of the food.
"Free from," on the other hand, seems to indicate more of an escape in the case of something that was more externally attached, as in, "Now at last I am free from the annoying attention of my mother, who wouldn't leave me alone until I moved to New York."
As for the last example, "rid of this terrible cold," I find "rid" to work better with an invader, such as a disease, or anything that is more "invasive" as opposed to "pervasive." In other words, I would say: "I need to get rid of that tick that's been biting me all day." I would not say: "I need to be free from that tick." And that's why I would use "rid of" with a cold, rather than "free" (no matter whether of or from).
Let me take it a bit further. If we extend the conceptualization to the word "freedom," I think we'll find more basis for differentiation in the choices between "free of" and "free from." So let's try a few examples.
"Freedom from want." "Freedom from fear." "Freedom from hunger." These phrases cannot be constructed using the word "of." They demonstrate of being free from an entity that is externally attached in a conceptually philosophical way; hunger besets you, fear comes upon you, "want" sinks its claws into you. If you can remove these things from your life, you are "free from" the undesirable attention (attack) of these things.
And one more distinction that I find to be quite telling:
"Free of" is best used with nouns ("At last I'm free of those thoughts), whereas "free from" works much better with verb forms ("At last I'm free from thinking that way). It just doesn't work well for me the other way around.
Best Answer
Ambiguity:
Amphibology:
Etymology: