And, in fact, wikimediafoundation.org says "nonprofit charitable organization". I don't know why people generally say "non-profit" instead of "charity", then - charity would be more precise and would probably be better perceived.
I'm afraid I disagree with you here. Non-profit vs for-profit is a distinction in taxation and precise. Charity vs not being charity may 1) no legal distinction in some cases and 2) Wikimedia Foundation could be no charity in some definition of non-US jurisdiction (and at worse it may be taken as deceitful).
I am for adding "charitable" etc. but against replacing "charity" etc. with "non-profit".
I would say "being charitable" and "being a charity" mean the same thing (in reference to an organisation). Under the UK definitions (I expect other jurisdictions are similar), a charity is a non-profit whose objects and activities fit the definition of charitable objects and activities (that definition may vary from place to place). Since the WMF is described as a charitable organisation on the official webpage, I assume it is correct to call it such, so "charity" is a more precise term than "non-profit". I don't think there is a jurisdictional problem - as long as it is a charity in its own jurisdiction, it should be fine to call it a charity on its own webpages.
The issue of varying cultural perceptions of the term "charity" (or literal translations) is a more serious one - we should give translators sufficient leeway to deal with such localisation issues.