I don't believe the expression "to talk impromptu" is a correct use of the Latin term.
You can make an impromptu speech, an impromptu remark, do an impromptu performance. The word impromptu functions more like an adjective.
The expression "off the cuff" is a phrase that stands on its own grammatically. It comes from the difference of reading "off the paper" of the prepared speech or "reading off the cuff" without a prepared speech. The cuff is the end of each arm of your shirt. Literally, you extend your arms out to the audience in a body language sense, and talk off the cuff ends of your shirt, instead of holding a piece of paper in your hand. The location word "off" means "from". "He fired off the gun", is another example.
So yes, "off the cuff" means literally without preparation. It can also mean without thinking.
You cannot really "lie off the cuff" because you are thinking that what you are saying is a lie. If your remark happens to be "off the cuff", you can later find out that it is wrong. That does not mean you lied off the cuff though. You instead said something without thinking if it is correct. If you are good at lying without preparation, then you are fluent in lying. You need a good memory though to remember what lies you have told, and so next time you must remember what you said to be consistent, otherwise you could contradict what you said last time. This needs preparation before speaking, even if you are very quick at preparing.