"In" is the preposition you use to say that one thing is CONTAINED inside another: "the water is IN the glass". It is very important that a bicyclist remain inside his lane. If he leaves his lane he might be hit by a car or another bicyclist. Since CONTAINMENT is of paramount importance, "IN" (or "within") is best preposition to use.

In the US, it would be "in the bike lane."

IN the cycle lane refers to being in that lane rather than another one or somewhere else on the road, so being in the cycle lane as opposed to in the bus lane or any other lane. ON the cycle lane is more focused on something being physically on that lane (on the surface) that should not be on any lane, like debris. So the text here is emphasizing being in the cycle lane rather than on a lane for everyone or only cars. Checking to see what people use more, I see that “on the cycle lane” is being used a little more in the UK and Ireland, whereas “in the cycle lane” is slightly more popular in the USA, Canada, Australia, and India. But the difference is not huge and could be due to other factors such as what people talk about being in/on the cycle lane.

In the cycle lane means that you are keeping within the borders of the lane. Another example of people being in a lane is swimmers who compete in a swimming race. They have to stay in their lanes.

You are always IN a lane, but ON the road. The difference lies in position....the lane markers are next to you, so you are in their boundaries, but you are on top of a road, or on a road.