I guess lane_markings is the correct one in this case... 
A group from Berlin put quite some effort to develop some detailed cycleway tagging scheme, but unfortunately it is written in German:
https://wiki.openstreetmap.org/wiki/Berlin/Verkehrswende/Radwege

There's also the draft of a proposal:
https://wiki.openstreetmap.org/wiki/Proposal:Separation#Symbolic_separation:_marking

I think this scheme is quite good, just needs some finalizing.