Seborrheic Dermatitis: This condition is characterized by a flaky, itchy, and red scalp, which matches the symptoms described. It's common and often confused with psoriasis, which could explain why psoriasis shampoo is not effective.

Contact Dermatitis: An allergic or irritant reaction to an ingredient in the shampoo or another hair product could cause a similar presentation, including redness, itching, and flakiness.
Psoriasis (not responding to treatment): Although the shampoo is not working, psoriasis itself is still a possibility, especially if the diagnosis was made based on clinical appearance alone without a biopsy. Resistance to treatment or incorrect use of the shampoo could be factors.
Fungal Infections (e.g., Ringworm of the Scalp): Tinea capitis can cause itchy, red, and flaky lesions on the scalp. It's more common in children but can occur in adults as well.

Cutaneous T-Cell Lymphoma (Mycosis Fungoides): Although rare, this condition can present with skin lesions that are itchy and can mimic more benign conditions. Missing this diagnosis could lead to delayed treatment of a potentially serious disease.
Sarcoidosis: This systemic disease can cause skin lesions, including on the scalp, and can be mistaken for other dermatological conditions. It's crucial to consider sarcoidosis due to its potential systemic implications.

Pityriasis Amiantacea: A condition characterized by the accumulation of scale around the hair shaft, leading to a thick, yellowish crust. It's often associated with psoriasis or seborrheic dermatitis but can be a distinct entity.
Keratosis Follicularis (Darier's Disease): A rare genetic disorder that affects the skin and can cause lesions on the scalp, among other areas. It's characterized by greasy, crusted papules.
Lichen Planopilaris: An inflammatory condition that leads to hair loss and can cause redness and itching on the scalp. It's part of a spectrum of lichen planus, which can affect the skin, mucous membranes, nails, and hair.