bicycle/pedestrian way can be tagged as

highway=path
bicycle=designated
foot=designated

OR 

highway=cycleway
foot=designated
(bicycle=designated is optional as highway=cycleway implies it)

I, and many cyclists-mappers, prefer the first one, but the second one is correct too. If foot=designated is added, then it's not considered a bicycle-only way.

On the other hand

highway=footway
bicycle=designated

Is not documented in the wiki and is not well interpreted by data consumers.

Ignore what iD editor tells you, iD editor is not a source of truth. For example, I've heard that Vespucci editor uses highway=path for such paths. CyclOSM map also prefers highway=path when both cyclists are pedestrians are allowed on it.

I'm not sure what you wanted to say by "converts". Perhaps you meant iD editor's template names or red/blue colours on the map?