From what I understand, a "simmer" is when you start getting small air bubbles floating up, but only one air bubble breaks through the liquid's surface every 2 seconds.
So then what is a "lively simmer" as I've seen in many recipes?
If I were to guess, it would be that the ones that call for a lively simmer are sauces, soups, or other thicker liquids. The lively simmer in that case would refer something close to a light boil. For instance, tomato sauce simmers even when you have it on low, but if you were to cut the temperature up to a medium low, it would bubble much more actively and to me would be considered a lively simmer.