All three of those words have meant many things and there is overlap in all 3. With what I think you mean, I agree. What a liberal is in the original sense of the word, a classical liberal, is what I think so called libertarians attempt to emulate while a libertarian originally meant a left-wing, anticapitalist anarchist. Republican is a term in use possibly earlier than the Roman empire in some form. In Rome, a republican was someone who opposed an imperial form of gov't but simply became someone who supported certain senators.
Words as we see, can be appropriated.