I'm not going to make any edits here now, since I also don't know if the bus lane should be mapped as a separate way. It's part of the same road, but based on your photos, it is separated from other traffic by bollards in this particular section of the road. I've added a bollard row based on your photos.

On a related note, I found on the wiki a proposal about separation, but it doesn't look like it's been touched in a few years. https://wiki.openstreetmap.org/wiki/Proposal:Separation