It is designated because from what I can tell it is a pedestrian and foot path and thus the tags used are highway=cycleway + foot=designated + bicycle=designated, those are just the community decided on tags for pedestrian and foot paths, that is why there is the general access=private because unless something specifically overrides the general access=* tag it applies to all uses, and the designated value doesn’t override the access=private because designated tells you whether a specific mode of transportation is allowed on a way and if so how, it does not tell you what the access requirements are (what private tells you). In essence the access private tells you that bicycle access is private and foot=designated + bicycle=designated tells you that only bikes and pedestrians are allowed to use the path which because of access=private means that only private pedestrians and bikes are allowed to use the path. I have already explained this in comments left on your commits previously. 