
Hi, Baloo Uriza!
My name is Manju and I’m a mapper in the OSM team at Lyft.
Based on the wiki, we don’t include bike lanes in the lane count because the highway=* key is mainly used by motorized traffic.
https://wiki.openstreetmap.org/wiki/Lanes#Crossing_with_a_designated_lane_for_bicycles
However, based on the other article, your enhanced approach to tagging lanes is also applicable.
https://wiki.openstreetmap.org/wiki/Key:lanes#:~:text=lane%20for%20those.-,NOTE,-%3A%20Not%20settled%3B%20orthogonal
Given our understanding, both approaches can be used.
Regarding turn lanes, according to our policy: https://github.com/Test-DCT/OSM-LYFT-DCT/blob/61f7554424aefbada1fde6d22fd71523c5f99d94/osmtrainingmaterialsfeatures/Turn%20Lanes.pdf, we add turn lane tags based on markings and signs. We do not include potential maneuvers, as in this case where you added the value "through;right"
Here, I have updated the data based on the street view from May 24th: https://drive.google.com/file/d/1aV4mfuu7bYGlbXqJIEiBntyp5-9goeyQ/view?usp=drive_link: 
Please let me know if you have any concern about that. Happy mapping! 

Best regards, Manju
