All of the above is true, but like anything else, it's complicated.
The base definition has to do with water chemistry. Limestone streams flow over or through limestone bedrock, which is water soluble. Thus, the water itself has a very high mineral content, and typically a very high alkalinity. Alkalinity is not pH, rather, it is the resistance to change of the pH. Spring water is typically on the basic side of the pH range, and thus, limestone streams will typically be basic and consistent in their pH.
Also, because limestone is soluble, caves form easily, and streams often flow underground for long distances. This results in LARGE springs in the deepest valleys. With any stream, the closer you are to the springs, the cooler it'll be in the summer, the warmer in the winter. In a limestoner, a sizable stream can be very close to its source springs, and temperature can be more consistent year round. That said, after emergence, they too flow above ground, and as you get farther downstream from the springs the temperature can start to vary like a freestoner.
It is a myth that freestoners are solely composed of runoff. Both limestoners and freestoners have source springs, and runoff can affect both.
However, freestoners flow over non-soluble bedrock. Thus, groundwater runs through soil, not bedrock. And the result is that the springs are much smaller and more numerous, typically higher on the mountainsides instead of down in the valley. This is why smaller freestoners are generally faster/higher gradient, and it's also why freestoners are more prone to a boom/bust flow; the time between rain and coming out in a spring is greatly reduced. But the easy (and proper) way to think of it is that in freestoners, the stream is on the surface much farther up into the headwaters. In limestoners, most of the headwaters are often underground, and the whole stream emerges as one somewhere downstream.
As far as water chemistry, freestoners have far fewer minerals and lower alkalinity. pH can be anywhere from basic to acidic, but it changes easily, so a stream may run basic but then a rainstorm will cause an acid spike. Streams start small and grow slowly, so they travel a long distance before becoming sizable. And because of that, a lower % of the flow at any given point is near to its source springs, causing temperature to vary more. Such streams usually have a "size" cutoff, where streams smaller than a certain size stay cool enough to support trout, but above a certain size warm too much in the summertime for wild trout. In PA, rarely do larger freestoners stay cool enough year-round, thus wild trout fishing is confined to smaller waters.
Sorry if that was too in depth....