I have found where you have changed path to cycleway a couple of times. This conversation began after I discovered this edit (while talking to someone cycling The Great Lake to Lake Trail). It is, however, 11 months old and things may have changed since then. 
https://www.openstreetmap.org/way/813680569  

The source of truth for all things OpenStreetMap (and the mechanism used for determining future truths) is the Wiki. From what I have gathered reading the wiki is "footway" and "cycleway" pre-date "path" by many years. They likely came from British legal terms. As OpenStreetMap matured the access and path tags were added. The big different between footway, cycleway, and path is their implied access values. 

* footway is foot=dedicated 
* cycleway is bicycle=dedicated 
* path is motor_vehicle=no 

From https://wiki.openstreetmap.org/wiki/Key:access : 
    "Access values pre-eminently describe legal permissions/restrictions and should follow ground truth, such as signage or legal ruling, and not involve guesswork. It does not describe common or typical use, even if the signage is generally ignored." 

In Michigan, we primarily use the "Signposted foot and bicycle path." example found here: 
https://wiki.openstreetmap.org/wiki/Tag:highway%3Dpath 

See the Michigan wiki for further explanation. If that page isn't clear, let me know and I will try to improve it.
https://wiki.openstreetmap.org/wiki/Michigan#Pedestrian_ways  