Copying my note reply here:

"""
I have mapped it as a dedicated highway=busway in changeset 161145087. Sadly, Carto does not render this; people've tried to get the tag supported for some years now. (An alternative tagging scheme exists by creating a sub-type of service road, but it's discouraged.)

However, other maps styles DO render dedicated busways, including the (imho superior) Americana style (americanamap.org). It can take a little while for new data to show up over there, though. Rest assured, it's in the database ;)

See the wiki: https://wiki.openstreetmap.org/wiki/Tag:highway%3Dbusway#Rendering

Feel free to reactivate this note or comment on my changeset. I'm always open to discussions. Cheers!
"""