What do you call something that is hung like a painting, but it is actually made by carving?

The usual word for something artistic that is hung is a painting; this covers the usual situation. But what I want to refer to is a hung wood piece. It has writings carved on it, and is painted.

Is there a word to describe any kind of work of art in a hung frame, rather than the usual (painting, drawing, picture)?

I found some pictures that would go through the category I'm looking for, like this or this.

A relief might be what you want.

Relief is a sculptural technique where the sculpted elements remain attached to a solid background of the same material.

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I do not think there is a general term with quite the specificity you suggest. Wood panel would connote a (mainly) flat piece of wood that may be painted, carved, have moldings etc - but this term does not distinguish a simple wall finish from either a decorative piece or a work of art. Panels are often framed though a frame is not essential. A panel does imply some type of body or substrate that holds everything together. In traditional (European) door construction the "panel" is the parts of the door door inside the rails and stiles i.e. the "frame".

This is a general answer for the general population. There are however specialized language groups which have their own terms. So one task you have is to determine who your audience is.

In the European and American art world, art was traditionally displayed in frames. The art itseld could be canvas (stretched on another, hidden frame), but it could also literally be a wood panel or cardboard or metal etc. The modern era however has really made the literal "frame" obsolete. Many paintings today are simply hung and no longer framed; many objects once denigrated as decoration or craft are now considered art; works no longer have to be fabricated by the artist but can be found ready made. In the world of commercial art the real answer to your question of what to call such a work is, whatever the artist wants to call it - and they may use terms such as "paintings","installations", "assemblages", "pieces", "works", "memory boxes", "shadow boxes", etc.

However, the works you have shown may also have historical and cultural significance. In this context works similar to those in your photos are often called called "bas-reliefs on wood", "carved wood panels" or just "wood carvings". These are probably the most general terms, but an historian or archaeologist is likely to be even more specific - especially if he/ she knew the exact purpose of the object. Here the possibilities are almost endless, it could be a sign, a medallion, a triptych, a reliquary, an advertisement, etc.

Engraving, although it is usually meant etched metal. Wood engraving, perhaps?