To me, those define different sorts of objects (albeit with the same legal access rights).
- The “highway=footway” one says “this was built for use by pedestrians, and someone has added a segregated bicycle lane on the left (in the direction of travel of the way), which cyclists have a legal right to use”, but may be inferior to other “proper” cycle paths elsewhere.
- The “highway=path” one initially says nothing about whether it’s likely to be appropriate for cyclists, but the " bicycle=designated" modifier changes that to say “actually, this is a highway=cycleway” (i.e. not inferior to other proper cycle paths). It doesn’t say which side pedestrians or cyclists go.