This comes up often. Good post by Tordanik. A little history and information which will hopefully provide some insight:
In countries like the UK, specifically in England and Wales but not so much in Scotland, paths have different legal statuses. This is often true in other places of course. Originally in OSM highway=footway was used to represent public footpaths, a legally protected public right of way. But it was also used for canal towpaths, urban paths with no legal right of way and various other things. It was a workhorse tag.
A problem presents itself, however, if you want to do the traditional kind of Ordnance Survey-style map with different rendering styles for public rights of way. Technically speaking the Ordnance Survey include a note on their maps saying that the indication of a legal right of way on their maps should not be taken as proof of a right of way. But nearly always these renderings do indeed show a public footpath or other public right of way and as such are very useful for walking in the countryside.
To allow for this, and for routing, various tags were invented. access=yes/no/permissive and foot=yes along with every other imaginable *=yes/no/permissive for vehicles and animals etc. The trouble is that this doesn’t nearly provide for the complexities of the UK public rights of way, or other countries rights of way for that matter. Also these tags began to be used more as routing hints and notes on the de facto state of access and accessibility rather than as guidelines on legality of access.
So some UK mappers, including myself, began using highway=path for all those little paths that have no legal status plus the various access keys to provide routing hints and so forth.
Then a new tag was developed - designation. Some people aren’t keen on this at all but I like it a lot. It’s simple - one tag, one value and all the legal access rights become clear. Then the access keys are free to be what they have been mainly used for anyway: notes on accessibility and therefore routing hints.
However highway=path is now essentially defunct, because you can use highway=footway or highway=cycleway with an appropriate designation tag if it’s a legal right of way and various access keys, such as bicycle= or foot=permissive to record the de facto accessibility.
But the world is not going to end if you use highway=path. It’s just a rather semantically void tag.