You may be doing what I think you need to do, but I cannot tell from your description.
The piece of wood should be over the carpet AND over bare wood. A piece used as a threshold like that generally has a thick end that goes over the bare wood and a thinner end over the carpet. I do not see that a piece 3/8" thick would be sufficient for that.
The idea of this piece is to protect the carpet from wear. But while doing this it should also be unobtrusive. Go to a carpet store and see what they look like.
George