Atopic Dermatitis (Eczema): The presence of white patches on the tummy and back that become red when wet, along with the appearance of red, pimple-like erythema, is suggestive of atopic dermatitis. The fact that the infant is otherwise active and well, and has started solids, which can sometimes trigger or exacerbate eczema, supports this diagnosis.

Seborrhoeic Dermatitis: The description of white patches with a ring-like appearance and the presence of red erythematous lesions could also be indicative of seborrhoeic dermatitis, especially given the location on the abdomen and back.
Irritant Contact Dermatitis: The red rash on the upper lips could be due to irritant contact dermatitis, possibly from coming into contact with an irritant in food or other substances.
Allergic Contact Dermatitis: Similar to irritant contact dermatitis, an allergic reaction to something the infant has come into contact with could cause the skin rashes.

Congenital Syphilis: Although less likely, congenital syphilis can cause skin rashes and must be considered, especially if there's any concern about maternal syphilis status. It's crucial to rule out this condition due to its severe implications if left untreated.
Immunodeficiency: Certain immunodeficiencies can present with recurrent or severe skin infections or rashes. While rare, this is a critical diagnosis not to miss due to its significant impact on the child's health and management.

Langerhans Cell Histiocytosis: A rare condition that can cause skin rashes among other systemic symptoms. It's essential to consider this in the differential due to its potential severity and the need for specific treatment.
Incontinentia Pigmenti: A genetic disorder that affects the skin, hair, teeth, and central nervous system. It's characterized by skin lesions that evolve over time, which could initially present as rashes in infancy.