Hello and Welcome,
First if you haven’t already done so please read
The one that was rejected is a directional sign to various places some of which are cycle friendly. But there is no evidence that this is for a defined route for example a NCN route or a local named route that avoids roads etc. Directional signs as the clarification states are not acceptable.
The accepted one refers to the specific Workhouse cycleway and the Green wheel walk which is promoted by sustrans even although it is not an NCN. In that case as it is part of a named and defined route it is acceptable.
So the 2 are very different and in my view the appeal decision is correct in this case.
Perhaps worth following the Green wheel walk for more signs as it looks a good route and you can link to a website for proof.