I'm ashamed, so ashamed. I thought the HTML syntax for textareas was exactly the same as for other input fields... but it seems I was wrong. This works:
<textarea name="CUSTOM_FIELD_1\" cols=\"50\" rows=\"5\"
wrap=\"soft\" class=\"form-input\">{$CUSTOM_FIELD_1}</textarea>
This does not work:
<input type="textarea\" name=\"CUSTOM_FIELD_1\"
value =\"$CUSTOM_FIELD_1\" cols=\"50\" rows=\"5\" wrap=\"soft\" class=\"form-input\" />
I'm so sorry for wasting your time... It worked correctly right from the beginning. I just don't know how HTML works
