This is the correct tagging. The wiki cited in this discussion says that ways that are ALSO used as biketrails should be tagged as paths. This is not applicable to any segment I retagged: They are to be used EXCLUSIVELY as biketrails, any other access including foot is forbidden.

There is more: Currently, there is a discussion going on whether biketrails should rather be tagged as leisure=track or with a new leisure tag: https://community.openstreetmap.org/t/mountain-bike-downhill-pistes/132329 
The discussion is not yet finished, so I would wait before using this.

What is clear, is that highway=path is wrong for any of these segments. And besides, it is also essential for safety that it is clearly visible that these ways are different from the rest, so they are not accessed by unsuspecting pedestrians.