Port-Wine Stain (PWS): Given the description of the lesions being almost flat with capillaries and only slight elevation, PWS is a strong consideration. The size and location of the lesions, particularly the large one on the back, are consistent with PWS. PWS is a congenital vascular malformation that presents at birth and can vary greatly in size.

Congenital Hemangioma: Although the description leans more towards a flat lesion, congenital hemangiomas can present in various forms, including those that are not significantly elevated. The presence of multiple lesions could suggest this diagnosis, but the lack of significant elevation and the capillary nature of the lesions make PWS more likely.
Capillary Malformation-Arteriovenous Malformation (CM-AVM) Syndrome: This is a condition that could present with multiple vascular malformations, including those on the face and back. However, it typically involves more complex vascular anomalies and often includes other systemic features.

Sturge-Weber Syndrome (SWS): This syndrome is characterized by a port-wine stain on the face, typically in the distribution of the trigeminal nerve, along with neurological and ocular abnormalities. The presence of a large PWS, especially on the face, necessitates consideration of SWS to avoid missing associated brain and eye abnormalities.
Klippel-Trenaunay Syndrome (KTS): This condition involves congenital vascular malformations, often with limb hypertrophy and potential for other systemic vascular anomalies. While the primary description does not mention limb involvement, the large size of the back lesion and the presence of multiple lesions warrant consideration of KTS.

PHACE(S) Syndrome: This is a rare condition characterized by large facial hemangiomas and potential for cerebral, arterial, cardiac, eye, and sternal abnormalities. Although the description does not strongly suggest a hemangioma, the presence of multiple vascular malformations and their locations could prompt consideration of PHACE(S) syndrome in the differential diagnosis.
Cutis Marmorata Telangiectatica Congenita (CMTC): This is a rare condition presenting with reticulated vascular patterns and potential for associated anomalies. While it's less likely given the description of discrete lesions, it remains a consideration in the broad differential diagnosis of congenital vascular malformations.

To exclude other vascular connections of organs and associated syndromes, a comprehensive evaluation including:

Imaging Studies: Ultrasound, MRI, or MRA to assess the extent of the vascular malformations and potential involvement of internal organs.
Ophthalmologic Examination: Especially if there's a facial port-wine stain, to rule out glaucoma or other ocular abnormalities associated with Sturge-Weber Syndrome.
Neurological Evaluation: To assess for any signs of neurological impairment, particularly if Sturge-Weber Syndrome or other conditions with neurological involvement are suspected.
Cardiovascular Evaluation: To rule out any cardiac anomalies that might be associated with certain syndromes.
Genetic Counseling: For families with a history of similar conditions or when a genetic syndrome is suspected.

Early and thorough evaluation is crucial for managing congenital vascular malformations and associated conditions effectively.