As long as mtb’s are bicycles, I think bicycle access rules apply, with mtb=* as exception tagging. If mtb is seen as a type of bicycle that deserves its own access, I think a designated mtb-only path is bicycle=no + mtb=designated. In that case it does not matter much whether the highway is path or cycleway. Most mtb-tracks will be rough paths, I think.
highway=cycleway + bicycle=no + mtb=designated|yes requires some getting used to, but I think it is correct.
highway=path + bicycle=no + mtb=designated has the information.
You always have to take care to get foot and horse access right. Defaults about foot and horse access to paths and cycleways are country-dependent.
Story (no problem to skip this)
In Nederland, we require a legal cycleway sign to map cycleway, and a legal prohibition sign to map bicycle=no. For public traffic, we do not have official mtb-specific access and prohibition signs. So that is our problem; as usual, some mappers are strict and want to stick to legal access, and other mappers lean toward intent mapping rather than legal. I think mtb-enthousiasts are mostly in the second category…)
If a sign says (or anyone can tell from appearance) that a public path is meant for mtb’s, or mtb-only, it says nothing about legal access. The legal way would require traffic signs to exclude horse, foot and bicycle, with an undersign saying “Except mtb”.
On private land, the owner can use their own signage, which is legally binding as long as it clearly conveys the intention.
In practice, people don’t know all these legal and signage things, and just follow their instincts.
Another issue, both for cyclists and the law, is: what exactly is an mtb? THere are many variants of bicycles, and no strict descriptions of bikes that can use mtb-paths. And, no strict description of what is or is not an mtb path.