I don't think that this internal name has any legal binding, only the name of the WMF as a whole is legally bound (plus additional translations if they have been registered by the WMF as reserved brands). The generic legal term is independant of the existence of the WMF itself and would apply to other organizations. Locally, it may still have translations even if it works according to a legal definition. So it does not matter for example if we have "foundations", "associations" for non-profit organizations (but generally to benefit from tax-exemptions and accept donations, those organizations need some legal accounting and will be incorportated in the US
In Europe this is not necessary, associations work as non-profit and may accept donations and benefit from tax-exmpotion provided they have a reliable accounting, and most of them will have their accounting certified by an external auditor, generally a commerical company or certified professional; there are a few other requirements for tax-exemptions: tracability, social reponsabilities if they emply people, annual meeting, an elected "bureau", but membership is generally accessible to everyone, including corporates, and membership fees are payed depending on status of members but notamlly do not give excessive voting rights to corporates; if these corporate members have a majority of votes or someone controls more than 50% of votes, it can no longer be an association and the organization will have to become a corporation/company and the association dissolved.
"Foundations" in US are much more relaxed because only one member may have the full power on it and may provide almost all its working budget, other members only have a consultative right. These would be coprorations in Europe and would not be eligible to provide tax-emptions to donators, unless these donators are gourped in a separate affiliate where members have some decision and auditing power.
Anyway US is still not officially not a single-language country (English is not really an official language, it is just used "de facto" and many places in US accept other languages, notably French, Spanish, or native Amerindian languages in some areas). So various federal laws are effectively written origiannly in other languages (Spanish and French) and were translated only later (but in case of interpretation conflicts, the original versions are still binding and prevail). That's why so many contracts, and acts in US are explicitly stating the language in which it is origianally written, and explicitly assign a competent court in one specific state. The local jurisprudence has extablished the equivalences of terms on a case by case basis by interpreting the acts and contracts and just see if they were "fair", i.e. not abusive, and clear enough for all contractors when they were adopted/registered.
English is just used "de facto" at the federal level, but member states may have other official or working languages, and federal laws are not directly applicable in states without adoption by them (and states can still veto some laws voted at federal level, or change "legal terms"). Effectively the WMF is not really bound to US law but to the laws of the US state where it is registered and other states where it has some operations. What is then important is its act of incorporation in that state, whose language is significant and prevails in case to conflicts with other languages used and recognized in the same state.
But legal terms is one thing: for general communication, in documents that are not contractual or mandated by authorities or applicable laws, the WMF may still accept to have translations of its internal bodies and use them, because it will help better explain its activities to the world and to contributors. These internal boeies anyway are not trademarks and cannot be reserved for exclusive use by the WMF.