Are two-way center turn lanes included in “lanes=”?

Looks to me like there is a separate tag, see: https://wiki.openstreetmap.org/wiki/Key:centre_turn_lane

On the other hand, there is another section that says it should be counted as a lane: https://wiki.openstreetmap.org/wiki/Proposed_features/lanes_General_Extension/ProposalPreVoting#Center.2Fmedian_turn_lanes

There appears to be support for the "turn lanes" tagging in at least one renderer that I am aware of and in JOSM: https://wiki.openstreetmap.org/wiki/Key:turn#Turning_indications_per_lane

Given that mis-mash of tagging descriptions, I have avoided tagging center turn lanes. However I think I'd probably tag your example as:

lanes=5, lanes:forward=2, lanes:backward=2, lanes:both_ways=1, centre_turn_lane=yes

Which, I think, covers the different ways I see in the wiki. Looks like everyone else is confused too as there are only a small number of roads tagged either way even though that is a fairly common road geometry.