Wood etching and engraving seem to be similar techniques.
Wood printing is a different method.

Engraving and etching involve removing material from the thing being engraved or etched upon.

Printing is done by pressing something with an inked pattern on it onto paper or some other surface.

A piece of wood can be engraved - and then that engraved piece of wood can be used for printing.

Etching is generally a chemical process that removes material from metal or glass to leave a design on the thing being etched. Engraving can involved using hand tools to make the design by removing material from the surface.

and both woodcuts and wood engravings are printed on paper

Es gibt nur Holzschnitte und Hirnholzschnitte. Bei Holzschnitten wird das Material entfernt und die zu druckenden Teile stehen oberhalb der Fläche (Hochdruck). Bei Hirnholzschnitten wird das Bild seitenverkehrt ins Holz graviert (Intaglio), wie bei Kupfer- oder Stahlstichen.

When people talk about 'wood etching' they always seem to mean 'wood engraving'. Engraving is done with a tool, etching by definition with some corrosive chemical. I am not sure that wood etching actually exists as a technique.

Wood printing presumably means printing from wood blocks, as is done with woodcuts and Japanese woodblock prints.

The difference between wood engraving and woodcuts/woodblock printing is that in the former, the ink is in the lines cut into the surface by the engraving tool, in the latter, it is the lines or areas that stand proud of the surface that are inked (i.e. the lines or areas left behind after the remainder of the surface has been scraped away). .

Thank you very much for your help. You've been really kind.
The differences are clear now.