What connection is there between Mapnik, Cyclemap and the others?

1. They render different objects. Mapnik wants to be a general map, so renders everything that isn't too obstructive for the view (basically, everything but routes). Other maps try to offer a cleaner map.
2. They are rendered at different times. It has no purpose of rendering a tile when nobody looks at it. So if you constantly look at the Mapnik style, that one will be rendered again a lot if times, while the other styles won't be rendered again (until you look at it).
3. The renderers are hosted by different servers. Mapnik comes directly from OSM servers, Mapquest comes from Mapquest servers, and I believe that other map styles come from other servers. Some servers are just faster than others.

The renderers don't exchange data, but get it from the same source (the OSM database). So they could be equally fast.

A famous OSM guideline is "don't tag for the renderer". The renderer is great to see if you've done something wrong (like a typo in a tag), but if one renderer has it right, the data is probably good, so you shouldn't worry about the other renderers. They will follow.