Hi,

Your remapped highway=cycleway=lane is same problem as before, invalid mapping/tagging. If you like to see the cycleways already mapped and rendered visit the openstreetmap  website and select layers Y (CycleOSM) and C (Cycle Map) to see them. Sometimes it has a day delay before new ones show as the maps are not refreshed continuously.

As sample view. Cycleways tagged on a road are directly affixed to the side of the road and are dashed lines blue. Cycleways mapped separately will show as individual ways that follow their own way and are solid blue. An example
https://www.openstreetmap.org/changeset/160122232#map=20/42.7268387/13.9847495&layers=Y

Names generally go on the route relations, not on the road segments it self if directly parellel to the road. Here you see they carry the tag BI-6 since they are officially part of the Ciclovia Adriatica 6, also known as Bice Italia 6.

Ciao