In terms of OSM data there is no difference. What's saved in database are tags you see at the bottom when you select an object. iD editor's search and templates are just a convenient way to fill those tags. "Feature type" you see at the top is not saved and is not visible to applications consuming OSM data. It will be up to them to interpret data from the tags correctly.

The only alternative we have is to make cut line and the path separate lines drawn on top of each other.

Depends on if you consider them a same feature.
https://wiki.openstreetmap.org/wiki/One_feature,_one_OSM_element

I think both options are ok, though wiki says

If a road (usually highway=track or highway=path) follows the cutline, it is to be tagged as such in addition to cutline tags. 
https://wiki.openstreetmap.org/wiki/Tag:man_made%3Dcutline

