Aloha! You're speaking with Dr. Michael Salkin
I'm sorry for the delay in responding to you. We don't have many avian vets on the UK site just yet.
Welcome to the world of backyard chickens. To answer your easiest question first - yes, the areas on her body affected represent moulting - particularly because the other hen is definitely moulting and it's the time of year we expect moulting. The next question is more of a challenge. You've described the "sick chicken" which can be quite a challenge to address when no other symptoms other than lethargy and anorexia are present. Please let me know if you've noticed any respiratory symptoms - nasal discharge, conjunctivitis, gaping, sneezing, coughing - or diarrhea in this hen.
Nutritional, parasitic, and infectious disorders are all too common in our chickens. Unfortunately, chickens - like all birds - are often quite ill by the time they demonstrate symptoms. This is to their benefit because they'll be attacked by other chickens or predators if thought to be ill. It's not to our benefit because without a physical exam by an avian vet and perhaps biochemical testing and cultures, we can't know what the etiology of their illness is.
We know that moulting is a time of stress and need for additional protein, calcium and vitamin D3. If you're seeing the depression associated with a deficiency of these requirements, tube force-feeding of your hen would be appropriate. You'll need to be shown how this is done, however, by an avian vet in order to avoid the food going down the wrong pipe which would end up causing an aspiration pneumonia. For the time being, a slurry made from ground up pelleted feed/crumbles + a vitamin/mineral supplement available at your local feed store can be administered by dropper. The crop should be gently distended by the time you've finished force-feeding.
This vet in Taunton claims to have a special interest in birds:
Priory Close Veterinary Practice
Somerset, TA1 1SW
Tel: 01823 271042
I'm sorry I can't be more specific for you at this time. Such is the dilemma with many chicken disorders. Please respond with further questions or concerns if you wish.